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 .