Comprehensive Multiple-Choice Question Set: Rigorous Study of Real
Numbers and Real-Valued Functions
1. Which of the following is not an algebraic property of real numbers?
a) Associativity
b) Commutativity
c) Distributivity
d) Trichotomy
Answer: d) Trichotomy
2. What is the Archimedean property of real numbers?
a) For any real number x, there exists a natural number n such that n > x
b) Every non-empty set of real numbers that is bounded above has a least upper bound
c) Between any two distinct real numbers, there is always a rational number
d) The set of real numbers is uncountable
Answer: a) For any real number x, there exists a natural number n such that n > x
3. Which of the following best describes the Completeness Axiom of real numbers?
a) Every Cauchy sequence of real numbers converges
b) Every non-empty set of real numbers that is bounded above has a least upper bound
c) Between any two real numbers, there is another real number
d) The set of real numbers is dense in itself
Answer: b) Every non-empty set of real numbers that is bounded above has a least upper
bound
4. What is the cardinality of the set of real numbers?
a) (aleph-null)ℵ₀
b) (aleph-one)ℵ₁
c) 2^ℵ₀
d) Countably infinite
Answer: c) 2^ℵ₀
5. Which of the following is true about rational numbers?
a) They form a complete ordered field
b) They are dense in the real numbers
c) They are uncountable
d) They have the least upper bound property
Answer: b) They are dense in the real numbers
6. What is the definition of the limit of a function f(x) as x approaches a?
a) f(x) = L
b) For every ε > 0, there exists δ > 0 such that |f(x) - L| < ε whenever 0 < |x - a| < δ
c) f(x) is continuous at x = a
d) f(x) is differentiable at x = a
Answer: b) For every ε > 0, there exists δ > 0 such that |f(x) - L| < ε whenever 0 < |x - a| < δ
7. Which of the following is not a necessary condition for a function to be continuous at a point?
a) The function must be defined at that point
b) The limit of the function must exist as x approaches that point
c) The limit must equal the function value at that point
d) The function must be differentiable at that point
Answer: d) The function must be differentiable at that point
8. What is the Intermediate Value Theorem?
a) If f is continuous on [a,b] and f(a) < k < f(b), then there exists c in (a,b) such that f(c) = k
b) If f is differentiable on [a,b], then there exists c in (a,b) such that f'(c) = [f(b) - f(a)]/(b-a)
c) If f is continuous on [a,b], then f is bounded on [a,b]
d) If f is continuous on [a,b], then f attains its maximum and minimum values on [a,b]
Answer: a) If f is continuous on [a,b] and f(a) < k < f(b), then there exists c in (a,b) such that
f(c) = k
9. What is the definition of uniform continuity?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all x,y in
the domain
b) The function is continuous at every point in its domain
c) The function has a continuous derivative
d) The function is bounded on its domain
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all x,y in the domain
10. Which of the following is equivalent to the statement "f is differentiable at x = a"?
a) f is continuous at x = a
b) The limit of [f(x) - f(a)]/(x-a) exists as x approaches a
c) f has a local maximum or minimum at x = a
d) f is integrable at x = a
Answer: b) The limit of [f(x) - f(a)]/(x-a) exists as x approaches a
11. What is Rolle's Theorem?
a) If f is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), then there exists c in
(a,b) such that f'(c) = 0
b) If f is differentiable on [a,b], then f(b) - f(a) = f'(c)(b-a) for some c in (a,b)
c) If f is continuous on [a,b], then f is integrable on [a,b]
d) If f is differentiable on (a,b), then f is continuous on (a,b)
Answer: a) If f is continuous on [a,b] and differentiable on (a,b), and f(a) = f(b), then there
exists c in (a,b) such that f'(c) = 0
12. What is the Mean Value Theorem?
a) If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) such that
f'(c) = [f(b) - f(a)]/(b-a)
b) If f is integrable on [a,b], then there exists c in [a,b] such that ∫[a,b] f(x)dx = f(c)(b-a)
c) If f is continuous on [a,b], then the average value of f on [a,b] equals f(c) for some c in [a,b]
d) If f is differentiable on (a,b), then f'(x) is continuous on (a,b)
Answer: a) If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b)
such that f'(c) = [f(b) - f(a)]/(b-a)
13. Which of the following is not a consequence of the Mean Value Theorem?
a) If f'(x) = 0 for all x in (a,b), then f is constant on [a,b]
b) If f'(x) ≥ 0 for all x in (a,b), then f is increasing on [a,b]
c) If |f'(x)| ≤ M for all x in (a,b), then |f(b) - f(a)| ≤ M|b-a|
d) If f'(x) exists for all x in (a,b), then f is continuous on (a,b)
Answer: d) If f'(x) exists for all x in (a,b), then f is continuous on (a,b)
14. What is the definition of Riemann integrability on [a,b]?
a) The function must be continuous on [a,b]
b) The function must be bounded on [a,b]
c) The upper and lower Riemann sums must converge to the same value
d) The function must have an antiderivative on [a,b]
Answer: c) The upper and lower Riemann sums must converge to the same value
15. Which of the following is not a sufficient condition for Riemann integrability on [a,b]?
a) f is continuous on [a,b]
b) f is monotonic on [a,b]
c) f has finitely many discontinuities on [a,b]
d) f is differentiable on [a,b]
Answer: d) f is differentiable on [a,b]
16. What is the Fundamental Theorem of Calculus, Part I?
a) If f is continuous on [a,b], then F(x) = ∫[a,x] f(t)dt is differentiable on (a,b) and F'(x) = f(x)
b) If F'(x) = f(x) on [a,b], then ∫[a,b] f(x)dx = F(b) - F(a)
c) If f is integrable on [a,b], then f is continuous on [a,b]
d) If f is differentiable on [a,b], then f is integrable on [a,b]
Answer: a) If f is continuous on [a,b], then F(x) = ∫[a,x] f(t)dt is differentiable on (a,b) and F'(x)
= f(x)
17. What is a necessary and sufficient condition for a sequence of real numbers to converge?
a) It is bounded
b) It is monotonic
c) It is Cauchy
d) It has a convergent subsequence
Answer: c) It is Cauchy
18. Which of the following is true about every convergent sequence of real numbers?
a) It is monotonic
b) It is bounded
c) It is eventually constant
d) It has a unique limit
Answer: b) It is bounded
19. What is the Bolzano-Weierstrass Theorem?
a) Every bounded sequence has a convergent subsequence
b) Every Cauchy sequence converges
c) Every monotonic bounded sequence converges
d) Every convergent sequence is Cauchy
Answer: a) Every bounded sequence has a convergent subsequence
20. Which of the following is not a property of convergent series?
a) The sequence of partial sums converges
b) The general term approaches zero
c) The series remains convergent if finitely many terms are added or removed
d) The series converges absolutely
Answer: d) The series converges absolutely
21. What is the Ratio Test for series convergence?
a) If lim|a[n+1]/a[n]| < 1, the series converges
b) If lim|a[n+1]/a[n]| > 1, the series diverges
c) If lim|a[n+1]/a[n]| = 1, the test is inconclusive
d) All of the above
Answer: d) All of the above
22. Which of the following is true about power series?
a) They always converge for all real numbers
b) They have a radius of convergence
c) They are always uniformly convergent on their interval of convergence
d) They can only represent polynomial functions
Answer: b) They have a radius of convergence
23. What is the definition of uniform convergence for a sequence of functions {f[n]} to a function
f on a set E?
a) For every x in E, lim f[n](x) = f(x)
b) For every ε > 0, there exists N such that |f[n](x) - f(x)| < ε for all n ≥ N and all x in E
c) The sequence of derivatives {f[n]'} converges to f'
d) The sequence {f[n]} is equicontinuous on E
Answer: b) For every ε > 0, there exists N such that |f[n](x) - f(x)| < ε for all n ≥ N and all x in E
24. Which of the following is not preserved under uniform convergence?
a) Continuity
b) Boundedness
c) Integrability
d) Differentiability
Answer: d) Differentiability
25. What is the Stone-Weierstrass Theorem?
a) Every continuous function on a compact set can be uniformly approximated by polynomials
b) Every continuous function on R is the uniform limit of a sequence of step functions
c) Every bounded function on [a,b] is Riemann integrable
d) Every continuous function on [a,b] has an antiderivative
Answer: a) Every continuous function on a compact set can be uniformly approximated by
polynomials
26. Which of the following is an open set in the standard topology of R?
a) [0,1]
b) (0,1)
c) {0,1}
d) Q (the set of rational numbers)
Answer: b) (0,1)
27. What is the definition of a compact set in R?
a) It is closed and bounded
b) It is open and bounded
c) It contains all its limit points
d) It has a finite subcover for every open cover
Answer: a) It is closed and bounded
28. Which of the following is true about continuous functions on compact sets?
a) They are bounded
b) They attain their maximum and minimum values
c) They are uniformly continuous
d) All of the above
Answer: d) All of the above
29. What is the definition of a connected set in R?
a) It cannot be written as the union of two non-empty disjoint open sets
b) It is compact
c) It is bounded
d) It contains all its limit points
Answer: a) It cannot be written as the union of two non-empty disjoint open sets
30. Which of the following is an example of a nowhere dense set in R?
a) Q (the set of rational numbers)
b) [0,1]
c) (0,1)
d) The Cantor set
Answer: d) The Cantor set
31. What is the definition of a Cauchy sequence in R?
a) It converges to a real number
b) It is bounded
c) For every ε > 0, there exists N such that |a[m] - a[n]| < ε for all m,n ≥ N
d) It has a convergent subsequence
Answer: c) For every ε > 0, there exists N such that |a[m] - a[n]| < ε for all m,n ≥ N
32. Which of the following is equivalent to the completeness of R?
a) Every Cauchy sequence converges
b) Every bounded sequence has a convergent subsequence
c) Every non-empty set that is bounded above has a least upper bound
d) All of the above
Answer: d) All of the above
33. What is the definition of a limit point of a set E in R?
a) It is an element of E
b) Every neighborhood of it contains infinitely many points of E
c) It is the limit of a convergent sequence in E
d) It is an isolated point of E
Answer: b) Every neighborhood of it contains infinitely many points of E
34. Which of the following is true about closed sets in R?
a) They contain all their limit points
b) Their complement is open
c) They are compact if and only if they are bounded
d) All of the above
Answer: d) All of the above
35. What is the Heine-Borel Theorem?
a) A subset of R is compact if and only if it is closed and bounded
b) Every open cover of a compact set has a finite subcover
c) Every continuous function on a compact set is uniformly continuous
d) All of the above
Answer: d) All of the above
36. Which of the following is not a property of continuous functions?
a) The composition of continuous functions is continuous
b) The sum and product of continuous functions are continuous
c) The quotient of continuous functions is always continuous
d) The inverse of a continuous bijective function on a compact set is continuous
Answer: c) The quotient of continuous functions is always continuous
37. What is the definition of uniform continuity?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all x,y in
the domain
b) The function is continuous at every point in its domain
c) The function has a continuous derivative
d) The function is bounded on its domain
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all x,y in the domain
38. Which of the following is true about uniformly continuous functions?
a) Every continuous function on a compact set is uniformly continuous
b) The composition of uniformly continuous functions is uniformly continuous
c) If f is uniformly continuous on (a,b), then it can be extended to a continuous function on
[a,b]
d) All of the above
Answer: d) All of the above
39. What is the definition of absolute continuity for a function f on [a,b]?
a) f is continuous and differentiable on [a,b]
b) For every ε > 0, there exists δ > 0 such that Σ|f(b[k]) - f(a[k])| < ε for any finite collection of
disjoint subintervals (a[k],b[k]) with Σ(b[k]-a[k]) < δ
c) f has bounded variation on [a,b]
d) f is the indefinite integral of its derivative
Answer: b) For every ε > 0, there exists δ > 0 such that Σ|f(b[k]) - f(a[k])| < ε for any finite
collection of disjoint subintervals (a[k],b[k]) with Σ(b[k]-a[k]) < δ
40. Which of the following is true about absolutely continuous functions?
a) They are of bounded variation
b) They are differentiable almost everywhere
c) They satisfy the fundamental theorem of calculus
d) All of the above
Answer: d) All of the above
41. What is the definition of a function of bounded variation on [a,b]?
a) The function is bounded on [a,b]
b) The total variation of the function on [a,b] is finite
c) The function is continuous on [a,b]
d) The function is differentiable on [a,b]
Answer: b) The total variation of the function on [a,b] is finite
42. Which of the following is not necessarily true for functions of bounded variation?
a) They can be expressed as the difference of two increasing functions
b) They are continuous
c) They are differentiable almost everywhere
d) They have at most countably many discontinuities
Answer: b) They are continuous
43. What is the Lebesgue differentiation theorem?
a) Every continuous function is differentiable almost everywhere
b) The derivative of an absolutely continuous function is its Radon-Nikodym derivative
c) For almost every x, the average value of an integrable function over a small interval around
x converges to f(x) as the interval shrinks
d) The derivative of a monotonic function exists almost everywhere
Answer: c) For almost every x, the average value of an integrable function over a small
interval around x converges to f(x) as the interval shrinks
44. Which of the following is equivalent to Lebesgue integrability on [a,b]?
a) Riemann integrability
b) Continuity
c) The function is the pointwise limit of a sequence of simple functions
d) The function is bounded
Answer: c) The function is the pointwise limit of a sequence of simple functions
45. What is Fatou's Lemma?
a) lim inf ∫f[n] ≤ ∫lim inf f[n]
b) lim sup ∫f[n] ≥ ∫lim sup f[n]
c) If f[n] → f pointwise, then ∫f = lim ∫f[n]
d) If f[n] → f uniformly, then ∫f = lim ∫f[n]
Answer: a) lim inf ∫f[n] ≤ ∫lim inf f[n]
46. What is the Monotone Convergence Theorem?
a) If f[n] is an increasing sequence of non-negative measurable functions, then ∫lim f[n] = lim
∫f[n]
b) If f[n] → f pointwise and |f[n]| ≤ g for an integrable g, then ∫f = lim ∫f[n]
c) If f[n] → f uniformly, then ∫f = lim ∫f[n]
d) If f[n] is a Cauchy sequence in L¹, then it converges to an L¹ function
Answer: a) If f[n] is an increasing sequence of non-negative measurable functions, then ∫lim
f[n] = lim ∫f[n]
47. What is the Dominated Convergence Theorem?
a) If f[n] is an increasing sequence of non-negative measurable functions, then ∫lim f[n] = lim
∫f[n]
b) If f[n] → f pointwise and |f[n]| ≤ g for an integrable g, then ∫f = lim ∫f[n]
c) If f[n] → f uniformly, then ∫f = lim ∫f[n]
d) If f[n] is a Cauchy sequence in L¹, then it converges to an L¹ function
Answer: b) If f[n] → f pointwise and |f[n]| ≤ g for an integrable g, then ∫f = lim ∫f[n]
48. What is the definition of the L^p norm for 1 ≤ p < ∞?
a) (∫|f|^p)^(1/p)
b) ess sup |f|
c) ∫|f|
d) (∫|f|^p)^p
Answer: a) (∫|f|^p)^(1/p)
49. What is Hölder's inequality?
a) ∫|fg| ≤ (∫|f|^p)^(1/p) (∫|g|^q)^(1/q) where 1/p + 1/q = 1
b) ∫|f+g|^p ≤ 2^(p-1)(∫|f|^p + ∫|g|^p)
c) ||f+g||[p] ≤ ||f||[p] + ||g||[p]
d) ||fg||[1] ≤ ||f||[p] ||g||[q] where 1/p + 1/q = 1
Answer: a) ∫|fg| ≤ (∫|f|^p)^(1/p) (∫|g|^q)^(1/q) where 1/p + 1/q = 1
50. What is Minkowski's inequality?
a) ∫|fg| ≤ (∫|f|^p)^(1/p) (∫|g|^q)^(1/q) where 1/p + 1/q = 1
b) ∫|f+g|^p ≤ 2^(p-1)(∫|f|^p + ∫|g|^p)
c) ||f+g||[p] ≤ ||f||[p] + ||g||[p]
d) ||fg||[1] ≤ ||f||[p] ||g||[q] where 1/p + 1/q = 1
Answer: c) ||f+g||[p] ≤ ||f||[p] + ||g||[p]
51. What is the Riesz-Fischer Theorem?
a) L^p is complete for 1 ≤ p ≤ ∞
b) Every bounded linear functional on L^p has a unique representation as an integral
c) The dual of L^p is L^q where 1/p + 1/q = 1
d) Continuous functions are dense in L^p for 1 ≤ p < ∞
Answer: a) L^p is complete for 1 ≤ p ≤ ∞
52. What is the definition of a measure zero set?
a) A set that can be covered by a countable collection of intervals with arbitrarily small total
length
b) A set with no elements
c) A set with finite Lebesgue measure
d) A set whose complement has infinite measure
Answer: a) A set that can be covered by a countable collection of intervals with arbitrarily
small total length
53. What is the Cantor-Lebesgue function?
a) A continuous, increasing function that is constant almost everywhere
b) A nowhere differentiable continuous function
c) A function that maps the Cantor set onto [0,1]
d) A function that is discontinuous at every rational point
Answer: a) A continuous, increasing function that is constant almost everywhere
54. What is the Baire Category Theorem?
a) The intersection of countably many dense open sets in a complete metric space is dense
b) Every complete metric space is a Baire space
c) A complete metric space cannot be written as the countable union of nowhere dense sets
d) All of the above
Answer: d) All of the above
55. What is the definition of a nowhere dense set?
a) A set whose closure has empty interior
b) A set with measure zero
c) A set that is not dense in any open interval
d) A set that contains no open intervals
Answer: a) A set whose closure has empty interior
56. Which of the following is an example of a perfect set?
a) Q (the set of rational numbers)
b) The Cantor set
c) (0,1)
d) {0,1,2,3,...}
Answer: b) The Cantor set
57. What is the definition of a Gδ set?
a) The countable intersection of open sets
b) The countable union of closed sets
c) The complement of a Fσ set
d) Both a and c
Answer: d) Both a and c
58. Which of the following is true about continuous functions on R?
a) The set of points where a continuous function is differentiable is a Gδ set
b) The set of points where a continuous function is not differentiable is an Fσ set
c) A continuous function is differentiable almost everywhere
d) Both a and b
Answer: d) Both a and b
59. What is the Baire class of a function?
a) The smallest ordinal α such that the function is the pointwise limit of a sequence of
functions of class < α
b) The number of times the function can be differentiated
c) The degree of the polynomial that best approximates the function
d) The number of discontinuities of the function
Answer: a) The smallest ordinal α such that the function is the pointwise limit of a sequence
of functions of class < α
60. What is the definition of a Lipschitz continuous function?
a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
b) f is differentiable with bounded derivative
c) f is uniformly continuous
d) f is absolutely continuous
Answer: a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
61. Which of the following is true about Lipschitz continuous functions?
a) They are uniformly continuous
b) They are absolutely continuous
c) They are differentiable almost everywhere
d) All of the above
Answer: d) All of the above
62. What is the definition of a contraction mapping on a metric space?
a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
b) f is continuous and the image of any bounded set is bounded
c) f is uniformly continuous
d) f is bijective and both f and f^(-1) are continuous
Answer: a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
63. What does the Contraction Mapping Theorem state?
a) Every contraction mapping on a complete metric space has a unique fixed point
b) Every continuous function on a compact metric space has a fixed point
c) Every isometry of a compact metric space has a fixed point
d) Every Lipschitz function on a complete metric space is uniformly continuous
Answer: a) Every contraction mapping on a complete metric space has a unique fixed point
64. What is the definition of a uniformly equicontinuous family of functions?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all f in
the family
b) Each function in the family is uniformly continuous
c) The family is bounded in the uniform norm
d) The family is compact in the topology of uniform convergence
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all f in the family
65. What is the Arzelà-Ascoli Theorem?
a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly bounded and
equicontinuous
b) Every uniformly convergent sequence of continuous functions converges to a continuous
function
c) Every pointwise convergent sequence of continuous functions converges to a measurable
function
d) Every sequence of differentiable functions has a uniformly convergent subsequence
Answer: a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly
bounded and equicontinuous
66. What is the definition of a convex function?
a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
b) f is differentiable and f' is increasing
c) f is twice differentiable and f'' ≥ 0
d) The graph of f lies below any of its tangent lines
Answer: a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
67. Which of the following is true about convex functions?
a) They are continuous on the interior of their domain
b) They have left and right derivatives at each point in the interior of their domain
c) If differentiable, their derivative is monotonically increasing
d) All of the above
Answer: d) All of the above
68. What is Jensen's inequality?
a) f(Σ[i=1 to n] λ[i]x[i]) ≤ Σ[i=1 to n] λ[i]f(x[i]) for λ[i] ≥ 0 with Σλ[i] = 1
b) |f(x) - f(y)| ≤ K|x - y| for some constant K
c) f(E[X]) ≤ E[f(X)] for any random variable X
d) Both a and c
Answer: d) Both a and c
52. What is the definition of a measure zero set?
a) A set that can be covered by a countable collection of intervals with arbitrarily small total
length
b) A set with no elements
c) A set with finite Lebesgue measure
d) A set whose complement has infinite measure
Answer: a) A set that can be covered by a countable collection of intervals with arbitrarily
small total length
53. What is the Cantor-Lebesgue function?
a) A continuous, increasing function that is constant almost everywhere
b) A nowhere differentiable continuous function
c) A function that maps the Cantor set onto [0,1]
d) A function that is discontinuous at every rational point
Answer: a) A continuous, increasing function that is constant almost everywhere
54. What is the Baire Category Theorem?
a) The intersection of countably many dense open sets in a complete metric space is dense
b) Every complete metric space is a Baire space
c) A complete metric space cannot be written as the countable union of nowhere dense sets
d) All of the above
Answer: d) All of the above
55. What is the definition of a nowhere dense set?
a) A set whose closure has empty interior
b) A set with measure zero
c) A set that is not dense in any open interval
d) A set that contains no open intervals
Answer: a) A set whose closure has empty interior
56. Which of the following is an example of a perfect set?
a) Q (the set of rational numbers)
b) The Cantor set
c) (0,1)
d) {0,1,2,3,...}
Answer: b) The Cantor set
57. What is the definition of a Gδ set?
a) The countable intersection of open sets
b) The countable union of closed sets
c) The complement of a Fσ set
d) Both a and c
Answer: d) Both a and c
58. Which of the following is true about continuous functions on R?
a) The set of points where a continuous function is differentiable is a Gδ set
b) The set of points where a continuous function is not differentiable is an Fσ set
c) A continuous function is differentiable almost everywhere
d) Both a and b
Answer: d) Both a and b
59. What is the Baire class of a function?
a) The smallest ordinal α such that the function is the pointwise limit of a sequence of
functions of class < α
b) The number of times the function can be differentiated
c) The degree of the polynomial that best approximates the function
d) The number of discontinuities of the function
Answer: a) The smallest ordinal α such that the function is the pointwise limit of a sequence
of functions of class < α
60. What is the definition of a Lipschitz continuous function?
a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
b) f is differentiable with bounded derivative
c) f is uniformly continuous
d) f is absolutely continuous
Answer: a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
61. Which of the following is true about Lipschitz continuous functions?
a) They are uniformly continuous
b) They are absolutely continuous
c) They are differentiable almost everywhere
d) All of the above
Answer: d) All of the above
62. What is the definition of a contraction mapping on a metric space?
a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
b) f is continuous and the image of any bounded set is bounded
c) f is uniformly continuous
d) f is bijective and both f and f^(-1) are continuous
Answer: a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
63. What does the Contraction Mapping Theorem state?
a) Every contraction mapping on a complete metric space has a unique fixed point
b) Every continuous function on a compact metric space has a fixed point
c) Every isometry of a compact metric space has a fixed point
d) Every Lipschitz function on a complete metric space is uniformly continuous
Answer: a) Every contraction mapping on a complete metric space has a unique fixed point
64. What is the definition of a uniformly equicontinuous family of functions?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all f in
the family
b) Each function in the family is uniformly continuous
c) The family is bounded in the uniform norm
d) The family is compact in the topology of uniform convergence
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all f in the family
65. What is the Arzelà-Ascoli Theorem?
a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly bounded and
equicontinuous
b) Every uniformly convergent sequence of continuous functions converges to a continuous
function
c) Every pointwise convergent sequence of continuous functions converges to a measurable
function
d) Every sequence of differentiable functions has a uniformly convergent subsequence
Answer: a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly
bounded and equicontinuous
66. What is the definition of a convex function?
a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
b) f is differentiable and f' is increasing
c) f is twice differentiable and f'' ≥ 0
d) The graph of f lies below any of its tangent lines
Answer: a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
67. Which of the following is true about convex functions?
a) They are continuous on the interior of their domain
b) They have left and right derivatives at each point in the interior of their domain
c) If differentiable, their derivative is monotonically increasing
d) All of the above
Answer: d) All of the above
68. What is Jensen's inequality?
a) f(Σ[i=1 to n] λ[i]x[i]) ≤ Σ[i=1 to n] λ[i]f(x[i]) for λ[i] ≥ 0 with Σλ[i] = 1
b) |f(x) - f(y)| ≤ K|x - y| for some constant K
c) f(E[X]) ≤ E[f(X)] for any random variable X
d) Both a and c
Answer: d) Both a and c
52. What is the definition of a measure zero set?
a) A set that can be covered by a countable collection of intervals with arbitrarily small total
length
b) A set with no elements
c) A set with finite Lebesgue measure
d) A set whose complement has infinite measure
Answer: a) A set that can be covered by a countable collection of intervals with arbitrarily
small total length
53. What is the Cantor-Lebesgue function?
a) A continuous, increasing function that is constant almost everywhere
b) A nowhere differentiable continuous function
c) A function that maps the Cantor set onto [0,1]
d) A function that is discontinuous at every rational point
Answer: a) A continuous, increasing function that is constant almost everywhere
54. What is the Baire Category Theorem?
a) The intersection of countably many dense open sets in a complete metric space is dense
b) Every complete metric space is a Baire space
c) A complete metric space cannot be written as the countable union of nowhere dense sets
d) All of the above
Answer: d) All of the above
55. What is the definition of a nowhere dense set?
a) A set whose closure has empty interior
b) A set with measure zero
c) A set that is not dense in any open interval
d) A set that contains no open intervals
Answer: a) A set whose closure has empty interior
56. Which of the following is an example of a perfect set?
a) Q (the set of rational numbers)
b) The Cantor set
c) (0,1)
d) {0,1,2,3,...}
Answer: b) The Cantor set
57. What is the definition of a Gδ set?
a) The countable intersection of open sets
b) The countable union of closed sets
c) The complement of a Fσ set
d) Both a and c
Answer: d) Both a and c
58. Which of the following is true about continuous functions on R?
a) The set of points where a continuous function is differentiable is a Gδ set
b) The set of points where a continuous function is not differentiable is an Fσ set
c) A continuous function is differentiable almost everywhere
d) Both a and b
Answer: d) Both a and b
59. What is the Baire class of a function?
a) The smallest ordinal α such that the function is the pointwise limit of a sequence of
functions of class < α
b) The number of times the function can be differentiated
c) The degree of the polynomial that best approximates the function
d) The number of discontinuities of the function
Answer: a) The smallest ordinal α such that the function is the pointwise limit of a sequence
of functions of class < α
60. What is the definition of a Lipschitz continuous function?
a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
b) f is differentiable with bounded derivative
c) f is uniformly continuous
d) f is absolutely continuous
Answer: a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
61. Which of the following is true about Lipschitz continuous functions?
a) They are uniformly continuous
b) They are absolutely continuous
c) They are differentiable almost everywhere
d) All of the above
Answer: d) All of the above
62. What is the definition of a contraction mapping on a metric space?
a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
b) f is continuous and the image of any bounded set is bounded
c) f is uniformly continuous
d) f is bijective and both f and f^(-1) are continuous
Answer: a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
63. What does the Contraction Mapping Theorem state?
a) Every contraction mapping on a complete metric space has a unique fixed point
b) Every continuous function on a compact metric space has a fixed point
c) Every isometry of a compact metric space has a fixed point
d) Every Lipschitz function on a complete metric space is uniformly continuous
Answer: a) Every contraction mapping on a complete metric space has a unique fixed point
64. What is the definition of a uniformly equicontinuous family of functions?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all f in
the family
b) Each function in the family is uniformly continuous
c) The family is bounded in the uniform norm
d) The family is compact in the topology of uniform convergence
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all f in the family
65. What is the Arzelà-Ascoli Theorem?
a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly bounded and
equicontinuous
b) Every uniformly convergent sequence of continuous functions converges to a continuous
function
c) Every pointwise convergent sequence of continuous functions converges to a measurable
function
d) Every sequence of differentiable functions has a uniformly convergent subsequence
Answer: a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly
bounded and equicontinuous
66. What is the definition of a convex function?
a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
b) f is differentiable and f' is increasing
c) f is twice differentiable and f'' ≥ 0
d) The graph of f lies below any of its tangent lines
Answer: a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
67. Which of the following is true about convex functions?
a) They are continuous on the interior of their domain
b) They have left and right derivatives at each point in the interior of their domain
c) If differentiable, their derivative is monotonically increasing
d) All of the above
Answer: d) All of the above
68. What is Jensen's inequality?
a) f(Σ[i=1 to n] λ[i]x[i]) ≤ Σ[i=1 to n] λ[i]f(x[i]) for λ[i] ≥ 0 with Σλ[i] = 1
b) |f(x) - f(y)| ≤ K|x - y| for some constant K
c) f(E[X]) ≤ E[f(X)] for any random variable X
d) Both a and c
Answer: d) Both a and c
52. What is the definition of a measure zero set?
a) A set that can be covered by a countable collection of intervals with arbitrarily small total
length
b) A set with no elements
c) A set with finite Lebesgue measure
d) A set whose complement has infinite measure
Answer: a) A set that can be covered by a countable collection of intervals with arbitrarily
small total length
53. What is the Cantor-Lebesgue function?
a) A continuous, increasing function that is constant almost everywhere
b) A nowhere differentiable continuous function
c) A function that maps the Cantor set onto [0,1]
d) A function that is discontinuous at every rational point
Answer: a) A continuous, increasing function that is constant almost everywhere
54. What is the Baire Category Theorem?
a) The intersection of countably many dense open sets in a complete metric space is dense
b) Every complete metric space is a Baire space
c) A complete metric space cannot be written as the countable union of nowhere dense sets
d) All of the above
Answer: d) All of the above
55. What is the definition of a nowhere dense set?
a) A set whose closure has empty interior
b) A set with measure zero
c) A set that is not dense in any open interval
d) A set that contains no open intervals
Answer: a) A set whose closure has empty interior
56. Which of the following is an example of a perfect set?
a) Q (the set of rational numbers)
b) The Cantor set
c) (0,1)
d) {0,1,2,3,...}
Answer: b) The Cantor set
57. What is the definition of a Gδ set?
a) The countable intersection of open sets
b) The countable union of closed sets
c) The complement of a Fσ set
d) Both a and c
Answer: d) Both a and c
58. Which of the following is true about continuous functions on R?
a) The set of points where a continuous function is differentiable is a Gδ set
b) The set of points where a continuous function is not differentiable is an Fσ set
c) A continuous function is differentiable almost everywhere
d) Both a and b
Answer: d) Both a and b
59. What is the Baire class of a function?
a) The smallest ordinal α such that the function is the pointwise limit of a sequence of
functions of class < α
b) The number of times the function can be differentiated
c) The degree of the polynomial that best approximates the function
d) The number of discontinuities of the function
Answer: a) The smallest ordinal α such that the function is the pointwise limit of a sequence
of functions of class < α
60. What is the definition of a Lipschitz continuous function?
a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
b) f is differentiable with bounded derivative
c) f is uniformly continuous
d) f is absolutely continuous
Answer: a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
61. Which of the following is true about Lipschitz continuous functions?
a) They are uniformly continuous
b) They are absolutely continuous
c) They are differentiable almost everywhere
d) All of the above
Answer: d) All of the above
62. What is the definition of a contraction mapping on a metric space?
a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
b) f is continuous and the image of any bounded set is bounded
c) f is uniformly continuous
d) f is bijective and both f and f^(-1) are continuous
Answer: a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
63. What does the Contraction Mapping Theorem state?
a) Every contraction mapping on a complete metric space has a unique fixed point
b) Every continuous function on a compact metric space has a fixed point
c) Every isometry of a compact metric space has a fixed point
d) Every Lipschitz function on a complete metric space is uniformly continuous
Answer: a) Every contraction mapping on a complete metric space has a unique fixed point
64. What is the definition of a uniformly equicontinuous family of functions?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all f in
the family
b) Each function in the family is uniformly continuous
c) The family is bounded in the uniform norm
d) The family is compact in the topology of uniform convergence
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all f in the family
65. What is the Arzelà-Ascoli Theorem?
a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly bounded and
equicontinuous
b) Every uniformly convergent sequence of continuous functions converges to a continuous
function
c) Every pointwise convergent sequence of continuous functions converges to a measurable
function
d) Every sequence of differentiable functions has a uniformly convergent subsequence
Answer: a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly
bounded and equicontinuous
66. What is the definition of a convex function?
a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
b) f is differentiable and f' is increasing
c) f is twice differentiable and f'' ≥ 0
d) The graph of f lies below any of its tangent lines
Answer: a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
67. Which of the following is true about convex functions?
a) They are continuous on the interior of their domain
b) They have left and right derivatives at each point in the interior of their domain
c) If differentiable, their derivative is monotonically increasing
d) All of the above
Answer: d) All of the above
68. What is Jensen's inequality?
a) f(Σ[i=1 to n] λ[i]x[i]) ≤ Σ[i=1 to n] λ[i]f(x[i]) for λ[i] ≥ 0 with Σλ[i] = 1
b) |f(x) - f(y)| ≤ K|x - y| for some constant K
c) f(E[X]) ≤ E[f(X)] for any random variable X
d) Both a and c
Answer: d) Both a and c
52. What is the definition of a measure zero set?
a) A set that can be covered by a countable collection of intervals with arbitrarily small total
length
b) A set with no elements
c) A set with finite Lebesgue measure
d) A set whose complement has infinite measure
Answer: a) A set that can be covered by a countable collection of intervals with arbitrarily
small total length
53. What is the Cantor-Lebesgue function?
a) A continuous, increasing function that is constant almost everywhere
b) A nowhere differentiable continuous function
c) A function that maps the Cantor set onto [0,1]
d) A function that is discontinuous at every rational point
Answer: a) A continuous, increasing function that is constant almost everywhere
54. What is the Baire Category Theorem?
a) The intersection of countably many dense open sets in a complete metric space is dense
b) Every complete metric space is a Baire space
c) A complete metric space cannot be written as the countable union of nowhere dense sets
d) All of the above
Answer: d) All of the above
55. What is the definition of a nowhere dense set?
a) A set whose closure has empty interior
b) A set with measure zero
c) A set that is not dense in any open interval
d) A set that contains no open intervals
Answer: a) A set whose closure has empty interior
56. Which of the following is an example of a perfect set?
a) Q (the set of rational numbers)
b) The Cantor set
c) (0,1)
d) {0,1,2,3,...}
Answer: b) The Cantor set
57. What is the definition of a Gδ set?
a) The countable intersection of open sets
b) The countable union of closed sets
c) The complement of a Fσ set
d) Both a and c
Answer: d) Both a and c
58. Which of the following is true about continuous functions on R?
a) The set of points where a continuous function is differentiable is a Gδ set
b) The set of points where a continuous function is not differentiable is an Fσ set
c) A continuous function is differentiable almost everywhere
d) Both a and b
Answer: d) Both a and b
59. What is the Baire class of a function?
a) The smallest ordinal α such that the function is the pointwise limit of a sequence of
functions of class < α
b) The number of times the function can be differentiated
c) The degree of the polynomial that best approximates the function
d) The number of discontinuities of the function
Answer: a) The smallest ordinal α such that the function is the pointwise limit of a sequence
of functions of class < α
60. What is the definition of a Lipschitz continuous function?
a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
b) f is differentiable with bounded derivative
c) f is uniformly continuous
d) f is absolutely continuous
Answer: a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
61. Which of the following is true about Lipschitz continuous functions?
a) They are uniformly continuous
b) They are absolutely continuous
c) They are differentiable almost everywhere
d) All of the above
Answer: d) All of the above
62. What is the definition of a contraction mapping on a metric space?
a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
b) f is continuous and the image of any bounded set is bounded
c) f is uniformly continuous
d) f is bijective and both f and f^(-1) are continuous
Answer: a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
63. What does the Contraction Mapping Theorem state?
a) Every contraction mapping on a complete metric space has a unique fixed point
b) Every continuous function on a compact metric space has a fixed point
c) Every isometry of a compact metric space has a fixed point
d) Every Lipschitz function on a complete metric space is uniformly continuous
Answer: a) Every contraction mapping on a complete metric space has a unique fixed point
64. What is the definition of a uniformly equicontinuous family of functions?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all f in
the family
b) Each function in the family is uniformly continuous
c) The family is bounded in the uniform norm
d) The family is compact in the topology of uniform convergence
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all f in the family
65. What is the Arzelà-Ascoli Theorem?
a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly bounded and
equicontinuous
b) Every uniformly convergent sequence of continuous functions converges to a continuous
function
c) Every pointwise convergent sequence of continuous functions converges to a measurable
function
d) Every sequence of differentiable functions has a uniformly convergent subsequence
Answer: a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly
bounded and equicontinuous
66. What is the definition of a convex function?
a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
b) f is differentiable and f' is increasing
c) f is twice differentiable and f'' ≥ 0
d) The graph of f lies below any of its tangent lines
Answer: a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
67. Which of the following is true about convex functions?
a) They are continuous on the interior of their domain
b) They have left and right derivatives at each point in the interior of their domain
c) If differentiable, their derivative is monotonically increasing
d) All of the above
Answer: d) All of the above
68. What is Jensen's inequality?
a) f(Σ[i=1 to n] λ[i]x[i]) ≤ Σ[i=1 to n] λ[i]f(x[i]) for λ[i] ≥ 0 with Σλ[i] = 1
b) |f(x) - f(y)| ≤ K|x - y| for some constant K
c) f(E[X]) ≤ E[f(X)] for any random variable X
d) Both a and c
Answer: d) Both a and c
52. What is the definition of a measure zero set?
a) A set that can be covered by a countable collection of intervals with arbitrarily small total
length
b) A set with no elements
c) A set with finite Lebesgue measure
d) A set whose complement has infinite measure
Answer: a) A set that can be covered by a countable collection of intervals with arbitrarily
small total length
53. What is the Cantor-Lebesgue function?
a) A continuous, increasing function that is constant almost everywhere
b) A nowhere differentiable continuous function
c) A function that maps the Cantor set onto [0,1]
d) A function that is discontinuous at every rational point
Answer: a) A continuous, increasing function that is constant almost everywhere
54. What is the Baire Category Theorem?
a) The intersection of countably many dense open sets in a complete metric space is dense
b) Every complete metric space is a Baire space
c) A complete metric space cannot be written as the countable union of nowhere dense sets
d) All of the above
Answer: d) All of the above
55. What is the definition of a nowhere dense set?
a) A set whose closure has empty interior
b) A set with measure zero
c) A set that is not dense in any open interval
d) A set that contains no open intervals
Answer: a) A set whose closure has empty interior
56. Which of the following is an example of a perfect set?
a) Q (the set of rational numbers)
b) The Cantor set
c) (0,1)
d) {0,1,2,3,...}
Answer: b) The Cantor set
57. What is the definition of a Gδ set?
a) The countable intersection of open sets
b) The countable union of closed sets
c) The complement of a Fσ set
d) Both a and c
Answer: d) Both a and c
58. Which of the following is true about continuous functions on R?
a) The set of points where a continuous function is differentiable is a Gδ set
b) The set of points where a continuous function is not differentiable is an Fσ set
c) A continuous function is differentiable almost everywhere
d) Both a and b
Answer: d) Both a and b
59. What is the Baire class of a function?
a) The smallest ordinal α such that the function is the pointwise limit of a sequence of
functions of class < α
b) The number of times the function can be differentiated
c) The degree of the polynomial that best approximates the function
d) The number of discontinuities of the function
Answer: a) The smallest ordinal α such that the function is the pointwise limit of a sequence
of functions of class < α
60. What is the definition of a Lipschitz continuous function?
a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
b) f is differentiable with bounded derivative
c) f is uniformly continuous
d) f is absolutely continuous
Answer: a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
61. Which of the following is true about Lipschitz continuous functions?
a) They are uniformly continuous
b) They are absolutely continuous
c) They are differentiable almost everywhere
d) All of the above
Answer: d) All of the above
62. What is the definition of a contraction mapping on a metric space?
a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
b) f is continuous and the image of any bounded set is bounded
c) f is uniformly continuous
d) f is bijective and both f and f^(-1) are continuous
Answer: a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
63. What does the Contraction Mapping Theorem state?
a) Every contraction mapping on a complete metric space has a unique fixed point
b) Every continuous function on a compact metric space has a fixed point
c) Every isometry of a compact metric space has a fixed point
d) Every Lipschitz function on a complete metric space is uniformly continuous
Answer: a) Every contraction mapping on a complete metric space has a unique fixed point
64. What is the definition of a uniformly equicontinuous family of functions?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all f in
the family
b) Each function in the family is uniformly continuous
c) The family is bounded in the uniform norm
d) The family is compact in the topology of uniform convergence
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all f in the family
65. What is the Arzelà-Ascoli Theorem?
a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly bounded and
equicontinuous
b) Every uniformly convergent sequence of continuous functions converges to a continuous
function
c) Every pointwise convergent sequence of continuous functions converges to a measurable
function
d) Every sequence of differentiable functions has a uniformly convergent subsequence
Answer: a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly
bounded and equicontinuous
66. What is the definition of a convex function?
a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
b) f is differentiable and f' is increasing
c) f is twice differentiable and f'' ≥ 0
d) The graph of f lies below any of its tangent lines
Answer: a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
67. Which of the following is true about convex functions?
a) They are continuous on the interior of their domain
b) They have left and right derivatives at each point in the interior of their domain
c) If differentiable, their derivative is monotonically increasing
d) All of the above
Answer: d) All of the above
68. What is Jensen's inequality?
a) f(Σ[i=1 to n] λ[i]x[i]) ≤ Σ[i=1 to n] λ[i]f(x[i]) for λ[i] ≥ 0 with Σλ[i] = 1
b) |f(x) - f(y)| ≤ K|x - y| for some constant K
c) f(E[X]) ≤ E[f(X)] for any random variable X
d) Both a and c
Answer: d) Both a and c
52. What is the definition of a measure zero set?
a) A set that can be covered by a countable collection of intervals with arbitrarily small total
length
b) A set with no elements
c) A set with finite Lebesgue measure
d) A set whose complement has infinite measure
Answer: a) A set that can be covered by a countable collection of intervals with arbitrarily
small total length
53. What is the Cantor-Lebesgue function?
a) A continuous, increasing function that is constant almost everywhere
b) A nowhere differentiable continuous function
c) A function that maps the Cantor set onto [0,1]
d) A function that is discontinuous at every rational point
Answer: a) A continuous, increasing function that is constant almost everywhere
54. What is the Baire Category Theorem?
a) The intersection of countably many dense open sets in a complete metric space is dense
b) Every complete metric space is a Baire space
c) A complete metric space cannot be written as the countable union of nowhere dense sets
d) All of the above
Answer: d) All of the above
55. What is the definition of a nowhere dense set?
a) A set whose closure has empty interior
b) A set with measure zero
c) A set that is not dense in any open interval
d) A set that contains no open intervals
Answer: a) A set whose closure has empty interior
56. Which of the following is an example of a perfect set?
a) Q (the set of rational numbers)
b) The Cantor set
c) (0,1)
d) {0,1,2,3,...}
Answer: b) The Cantor set
57. What is the definition of a Gδ set?
a) The countable intersection of open sets
b) The countable union of closed sets
c) The complement of a Fσ set
d) Both a and c
Answer: d) Both a and c
58. Which of the following is true about continuous functions on R?
a) The set of points where a continuous function is differentiable is a Gδ set
b) The set of points where a continuous function is not differentiable is an Fσ set
c) A continuous function is differentiable almost everywhere
d) Both a and b
Answer: d) Both a and b
59. What is the Baire class of a function?
a) The smallest ordinal α such that the function is the pointwise limit of a sequence of
functions of class < α
b) The number of times the function can be differentiated
c) The degree of the polynomial that best approximates the function
d) The number of discontinuities of the function
Answer: a) The smallest ordinal α such that the function is the pointwise limit of a sequence
of functions of class < α
60. What is the definition of a Lipschitz continuous function?
a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
b) f is differentiable with bounded derivative
c) f is uniformly continuous
d) f is absolutely continuous
Answer: a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
61. Which of the following is true about Lipschitz continuous functions?
a) They are uniformly continuous
b) They are absolutely continuous
c) They are differentiable almost everywhere
d) All of the above
Answer: d) All of the above
62. What is the definition of a contraction mapping on a metric space?
a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
b) f is continuous and the image of any bounded set is bounded
c) f is uniformly continuous
d) f is bijective and both f and f^(-1) are continuous
Answer: a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
63. What does the Contraction Mapping Theorem state?
a) Every contraction mapping on a complete metric space has a unique fixed point
b) Every continuous function on a compact metric space has a fixed point
c) Every isometry of a compact metric space has a fixed point
d) Every Lipschitz function on a complete metric space is uniformly continuous
Answer: a) Every contraction mapping on a complete metric space has a unique fixed point
64. What is the definition of a uniformly equicontinuous family of functions?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all f in
the family
b) Each function in the family is uniformly continuous
c) The family is bounded in the uniform norm
d) The family is compact in the topology of uniform convergence
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all f in the family
65. What is the Arzelà-Ascoli Theorem?
a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly bounded and
equicontinuous
b) Every uniformly convergent sequence of continuous functions converges to a continuous
function
c) Every pointwise convergent sequence of continuous functions converges to a measurable
function
d) Every sequence of differentiable functions has a uniformly convergent subsequence
Answer: a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly
bounded and equicontinuous
66. What is the definition of a convex function?
a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
b) f is differentiable and f' is increasing
c) f is twice differentiable and f'' ≥ 0
d) The graph of f lies below any of its tangent lines
Answer: a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
67. Which of the following is true about convex functions?
a) They are continuous on the interior of their domain
b) They have left and right derivatives at each point in the interior of their domain
c) If differentiable, their derivative is monotonically increasing
d) All of the above
Answer: d) All of the above
68. What is Jensen's inequality?
a) f(Σ[i=1 to n] λ[i]x[i]) ≤ Σ[i=1 to n] λ[i]f(x[i]) for λ[i] ≥ 0 with Σλ[i] = 1
b) |f(x) - f(y)| ≤ K|x - y| for some constant K
c) f(E[X]) ≤ E[f(X)] for any random variable X
d) Both a and c
Answer: d) Both a and c
52. What is the definition of a measure zero set?
a) A set that can be covered by a countable collection of intervals with arbitrarily small total
length
b) A set with no elements
c) A set with finite Lebesgue measure
d) A set whose complement has infinite measure
Answer: a) A set that can be covered by a countable collection of intervals with arbitrarily
small total length
53. What is the Cantor-Lebesgue function?
a) A continuous, increasing function that is constant almost everywhere
b) A nowhere differentiable continuous function
c) A function that maps the Cantor set onto [0,1]
d) A function that is discontinuous at every rational point
Answer: a) A continuous, increasing function that is constant almost everywhere
54. What is the Baire Category Theorem?
a) The intersection of countably many dense open sets in a complete metric space is dense
b) Every complete metric space is a Baire space
c) A complete metric space cannot be written as the countable union of nowhere dense sets
d) All of the above
Answer: d) All of the above
55. What is the definition of a nowhere dense set?
a) A set whose closure has empty interior
b) A set with measure zero
c) A set that is not dense in any open interval
d) A set that contains no open intervals
Answer: a) A set whose closure has empty interior
56. Which of the following is an example of a perfect set?
a) Q (the set of rational numbers)
b) The Cantor set
c) (0,1)
d) {0,1,2,3,...}
Answer: b) The Cantor set
57. What is the definition of a Gδ set?
a) The countable intersection of open sets
b) The countable union of closed sets
c) The complement of a Fσ set
d) Both a and c
Answer: d) Both a and c
58. Which of the following is true about continuous functions on R?
a) The set of points where a continuous function is differentiable is a Gδ set
b) The set of points where a continuous function is not differentiable is an Fσ set
c) A continuous function is differentiable almost everywhere
d) Both a and b
Answer: d) Both a and b
59. What is the Baire class of a function?
a) The smallest ordinal α such that the function is the pointwise limit of a sequence of
functions of class < α
b) The number of times the function can be differentiated
c) The degree of the polynomial that best approximates the function
d) The number of discontinuities of the function
Answer: a) The smallest ordinal α such that the function is the pointwise limit of a sequence
of functions of class < α
60. What is the definition of a Lipschitz continuous function?
a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
b) f is differentiable with bounded derivative
c) f is uniformly continuous
d) f is absolutely continuous
Answer: a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
61. Which of the following is true about Lipschitz continuous functions?
a) They are uniformly continuous
b) They are absolutely continuous
c) They are differentiable almost everywhere
d) All of the above
Answer: d) All of the above
62. What is the definition of a contraction mapping on a metric space?
a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
b) f is continuous and the image of any bounded set is bounded
c) f is uniformly continuous
d) f is bijective and both f and f^(-1) are continuous
Answer: a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
63. What does the Contraction Mapping Theorem state?
a) Every contraction mapping on a complete metric space has a unique fixed point
b) Every continuous function on a compact metric space has a fixed point
c) Every isometry of a compact metric space has a fixed point
d) Every Lipschitz function on a complete metric space is uniformly continuous
Answer: a) Every contraction mapping on a complete metric space has a unique fixed point
64. What is the definition of a uniformly equicontinuous family of functions?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all f in
the family
b) Each function in the family is uniformly continuous
c) The family is bounded in the uniform norm
d) The family is compact in the topology of uniform convergence
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all f in the family
65. What is the Arzelà-Ascoli Theorem?
a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly bounded and
equicontinuous
b) Every uniformly convergent sequence of continuous functions converges to a continuous
function
c) Every pointwise convergent sequence of continuous functions converges to a measurable
function
d) Every sequence of differentiable functions has a uniformly convergent subsequence
Answer: a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly
bounded and equicontinuous
66. What is the definition of a convex function?
a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
b) f is differentiable and f' is increasing
c) f is twice differentiable and f'' ≥ 0
d) The graph of f lies below any of its tangent lines
Answer: a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
67. Which of the following is true about convex functions?
a) They are continuous on the interior of their domain
b) They have left and right derivatives at each point in the interior of their domain
c) If differentiable, their derivative is monotonically increasing
d) All of the above
Answer: d) All of the above
68. What is Jensen's inequality?
a) f(Σ[i=1 to n] λ[i]x[i]) ≤ Σ[i=1 to n] λ[i]f(x[i]) for λ[i] ≥ 0 with Σλ[i] = 1
b) |f(x) - f(y)| ≤ K|x - y| for some constant K
c) f(E[X]) ≤ E[f(X)] for any random variable X
d) Both a and c
Answer: d) Both a and c
52. What is the definition of a measure zero set?
a) A set that can be covered by a countable collection of intervals with arbitrarily small total
length
b) A set with no elements
c) A set with finite Lebesgue measure
d) A set whose complement has infinite measure
Answer: a) A set that can be covered by a countable collection of intervals with arbitrarily
small total length
53. What is the Cantor-Lebesgue function?
a) A continuous, increasing function that is constant almost everywhere
b) A nowhere differentiable continuous function
c) A function that maps the Cantor set onto [0,1]
d) A function that is discontinuous at every rational point
Answer: a) A continuous, increasing function that is constant almost everywhere
54. What is the Baire Category Theorem?
a) The intersection of countably many dense open sets in a complete metric space is dense
b) Every complete metric space is a Baire space
c) A complete metric space cannot be written as the countable union of nowhere dense sets
d) All of the above
Answer: d) All of the above
55. What is the definition of a nowhere dense set?
a) A set whose closure has empty interior
b) A set with measure zero
c) A set that is not dense in any open interval
d) A set that contains no open intervals
Answer: a) A set whose closure has empty interior
56. Which of the following is an example of a perfect set?
a) Q (the set of rational numbers)
b) The Cantor set
c) (0,1)
d) {0,1,2,3,...}
Answer: b) The Cantor set
57. What is the definition of a Gδ set?
a) The countable intersection of open sets
b) The countable union of closed sets
c) The complement of a Fσ set
d) Both a and c
Answer: d) Both a and c
58. Which of the following is true about continuous functions on R?
a) The set of points where a continuous function is differentiable is a Gδ set
b) The set of points where a continuous function is not differentiable is an Fσ set
c) A continuous function is differentiable almost everywhere
d) Both a and b
Answer: d) Both a and b
59. What is the Baire class of a function?
a) The smallest ordinal α such that the function is the pointwise limit of a sequence of
functions of class < α
b) The number of times the function can be differentiated
c) The degree of the polynomial that best approximates the function
d) The number of discontinuities of the function
Answer: a) The smallest ordinal α such that the function is the pointwise limit of a sequence
of functions of class < α
60. What is the definition of a Lipschitz continuous function?
a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
b) f is differentiable with bounded derivative
c) f is uniformly continuous
d) f is absolutely continuous
Answer: a) |f(x) - f(y)| ≤ K|x - y| for some constant K and all x, y
61. Which of the following is true about Lipschitz continuous functions?
a) They are uniformly continuous
b) They are absolutely continuous
c) They are differentiable almost everywhere
d) All of the above
Answer: d) All of the above
62. What is the definition of a contraction mapping on a metric space?
a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
b) f is continuous and the image of any bounded set is bounded
c) f is uniformly continuous
d) f is bijective and both f and f^(-1) are continuous
Answer: a) d(f(x),f(y)) ≤ kd(x,y) for some 0 ≤ k < 1 and all x, y
63. What does the Contraction Mapping Theorem state?
a) Every contraction mapping on a complete metric space has a unique fixed point
b) Every continuous function on a compact metric space has a fixed point
c) Every isometry of a compact metric space has a fixed point
d) Every Lipschitz function on a complete metric space is uniformly continuous
Answer: a) Every contraction mapping on a complete metric space has a unique fixed point
64. What is the definition of a uniformly equicontinuous family of functions?
a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for all f in
the family
b) Each function in the family is uniformly continuous
c) The family is bounded in the uniform norm
d) The family is compact in the topology of uniform convergence
Answer: a) For every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ for
all f in the family
65. What is the Arzelà-Ascoli Theorem?
a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly bounded and
equicontinuous
b) Every uniformly convergent sequence of continuous functions converges to a continuous
function
c) Every pointwise convergent sequence of continuous functions converges to a measurable
function
d) Every sequence of differentiable functions has a uniformly convergent subsequence
Answer: a) A family of functions is relatively compact in C[a,b] if and only if it is uniformly
bounded and equicontinuous
66. What is the definition of a convex function?
a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
b) f is differentiable and f' is increasing
c) f is twice differentiable and f'' ≥ 0
d) The graph of f lies below any of its tangent lines
Answer: a) f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) for all x, y and 0 ≤ t ≤ 1
67. Which of the following is true about convex functions?
a) They are continuous on the interior of their domain
b) They have left and right derivatives at each point in the interior of their domain
c) If differentiable, their derivative is monotonically increasing
d) All of the above
Answer: d) All of the above
68. What is Jensen's inequality?
a) f(Σ[i=1 to n] λ[i]x[i]) ≤ Σ[i=1 to n] λ[i]f(x[i]) for λ[i] ≥ 0 with Σλ[i] = 1
b) |f(x) - f(y)| ≤ K|x - y| for some constant K
c) f(E[X]) ≤ E[f(X)] for any random variable X
d) Both a and c
Answer: d) Both a and c
69. What is the definition of a semicontinuous function?
a) A function that is continuous from above or below at each point
b) A function that is continuous on a dense subset of its domain
