prooving limits and differentials
Arun1990
420-chp4-part2.pptxContinuous vs. Uniformly Continuous
A function is continuous at if
such that
is continuous on a set if is continuous at every
A function is uniformly continuous on if
such that
Theorem
If is uniformly continuous on , then is continuous on .
1
is uniformly continuous on is continuous on .
Whenever is compact,
is uniformly continuous on is continuous on .
Proof: Let . Choose /3. Thus,
and
Thus, is uniformly continuous on , which implies is continuous on
Example:
2
Uniformly Continuous examples
Proof: Let . Let . Choose . Thus,
and
Thus, is uniformly continuous on , which implies is continuous on
Example:
3
Uniformly Continuous examples
uniformly continuous
not
uniformly continuous
δ
δ
Prove that is not uniformly continuous on .
e.g. x^3 is NOT unif cont on R. P=n+1/n, Q=n (n suff. large)
e.g. 1/x^2 is NOT unif cont on (0,1]. P=1/n, Q=2/n (n suff. large)
4
Since the above must hold for every small another way to show non-uniform continuity is with sequences.
f is not
uniformly continuous
Example: 1/x is uniformly continuous on [a,b] for any 0<a<b, but not uniformly continuous on (0,1).
Consider the sequences x_n=1/2n and y_n=1/n. Notice that |1/x_n – 1/y_n| = n \ge 1=\varepsilon.
5
For last statement: try [-2,-1], [-1,0], and [0,2].
(x,y) points are (-2,-7), (-1,1), (0,-1), (2,13)
6
∀ε > 0 ∃δ > 0 such that ∀p, q ∈ E with | p − q |<δ ⇒| f ( p) − f (q) |<ε.
"e>0 $d>0 such that
"p,qÎE with |p-q|<dÞ|f(p)-f(q)|<e.
3
Prove that () is both continuous on [0,
1]
and uniformly continuous on [0,1].
fxx
=
3
Prove that () is both continuous on [,]
and uniformly continuous on [,].
fxxab
ab
=
To show f is not uniformly continuous on X , you should demonstrate that there is some ε> 0 such that no matter what > 0 you pick, you can find points p, q ∈ X such that | p − q | < and | f ( p) − f (q) | ≥ε.
∀ε> 0 ∃δ > 0 s.t. ∀p, q with | p − q |<δ ⇒ | f ( p) − f (q) | <ε
∀ε> 0( ) ∃δ > 0( ) ∀p, q( )(| p − q | <δ ⇒ | f ( p) − f (q) | <ε)
has negation
∃ε> 0( ) ∀δ > 0( ) ∃p, q( )(| p − q | <δ and | f ( p) − f (q) | ≥ε)
Prove f ( x) = x 3 + x 2 − 4 has a real zero.
Prove f(x)=x
3
+x
2
-4 has a real zero.
Prove f ( x) = x 3 + x 2 − 4 has a real zero in (0,2).
Prove f(x)=x
3
+x
2
-4 has a real zero in (0,2).
Prove f ( x) = 2x 3 + x 2 − 3x −1 has three real zeros.
Prove f(x)=2x
3
+x
2
-3x-1 has three real zeros.
