Math Real Analysis

MATH 3001 W21 A3

D ue o

n M arch

8 I

(1) Does fnlxl= cos " Cx ) sin

" (x) converge

uniformly on DE R ? And on Da = fo, The] ?

⑦ Let fnlxk n (YF -t) , D= [1, a) , a>I . Does fn converge uniformly ? Does th converge uniformly on [1, a) ?

③ Let fnlxk PIT fff , XER . Does fn converge uniformly ?

14) Appose fn→ F nmfrrmly on D, and each fn is continuous on D . Let Xu be a sequence in D Svt . xh

→x as h-1A , with AED.

Show that

tsjmfnlxn ) = FIN .

(5) Let fnlxl = on D= Coil] . Show that fh converges uniformly on D to a differentiable function , but fin does not convergeuniformlyon D.

MATH 3001 W21 S3

c) IfnCHIE thnx - cosxl . By periodsHy, we only need to

consider 0 EXIST . adz ahxcosx = agtx - ahh= I- Lah 'd . So XH aux . ask is Moran

'

ng ft) and decreasing l- I as follows : A

it i >X O I SIT 5T FI 21T

4 -4 -4 4

Hence the Max is at Thf : Shawn = ah Ig - costly = tg .

⇒ Ifn CHIE 2- h ⇒ fn Converges ht full-o , uniformly

on IR , and hence uniformly on any DCR .

(2) full) -o th . For X> I :

fishy n CIT- i ) = agm etnenx

- e

\ h-7N 4h

x'h- I = ethfhx ,

B.H . FIFA - ntzhexetnenx

=

- Ypg 's

= lnx . effy e then"

= en × .

Therefore fully → flxklnx poihtme on [1 , a) . Now checkfor nwifrrm convergence .

I fnlxl -flat In @ then" -t) - en x ) -

→ Mean value theorem :

f-(b)- flat = fyg) for

f -q

some § C- Calf]. Here a=gf= thnx, f-(shes

so .. eth" - I = e

's

for some BEF, th thx] . thnx

⇒ Ifnlxl -flail = thnx - e } - hrxle ( et- 1) lux .

Hence xfyp.my/fnlxI-fCx7/Efetnha-1jena--5o . So yes , fn→f Mnf. oh [ 1,9] fast .

Does fncawevgemnformlyonf.to) ? Well, if it does , then

the Amit function must still be Flxtlnx . New

YI , In F - it - hnxl § In In

-i ) - en n " / w

take the specific .

menu

value x-- nn

= In ( h-bin - t ) / h→A

→ a

Hence nhjm, gyp , Ifn by -flat -or and so fn does not

converge mnformly ar Cha ) .

⑦ Fix xeR . We know that fins EITI, exists , for example by the patio test. The Amit function folk II ¥, is also called the exponential function , f-Cx) = ex .

ftp.p/fn-ulM-tnlHf=fnp,p 'III, = -

Hence fn does not converge uniformly on R (by the Cauchy condition of uniform convergence) .

④ We know that F is continuous on D . Also ,

Ifn Hn) -FINI = Ifn Kul- FHM -1 Fkn )- FINI E Ifn Kw) -Fkn) / t IFkn) -MN ) ⇐

Yep, I fix ) -MN ) t IFHnl-FINI --

n→a hEyo as→0 as fit F Mnf. D F is continuous

at xED .

Eh

(5) najma =o . So the pahlmsehmit frmThai is flxfo .

Of came, f is differentiable on D .

Now ftncxk x ""

converges paintwise b- the function

g.HI= { O oExa

1 X= I .

Each of the fin are cmhhnom on D, but the limit functioning is not . Hence fh does not converge mnformly on D .