Question

1. (a) Prove that a closed subset of a compact set is compact. (b) Let a, b € R and f: R → R, x H ax + b. Prove that f is continuous. Is f uniformly continuous?

Answer 1

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le Thus KC (uva) UFC As we know k is k is compact every open cover have lome finite Rublover for k. Thus. frike subcollection & of which covere k. Thus Kc/uva) UFC there a DES where s is a s is a frite eet. Since FCKÇ'Uva) und DOS > FC (UVA) UFC LES >> FC UV acs thes F is compact

(b)

For some a and b in R, we have

f(x)= ax +b.

Want to prove f is a continuous function.

Let Eyo, such that S= E If a a 70. hal Coneider and euppore In-yl Ls, for ny EIR, Now 1}(m)-f(y)] - | anto-. (ay+b)] I an-cyl 1.(n-y)] Llallny) <lal.s.= lale = € 191 1 (n)-f(y)] LES Thus fes continuous IT le

Here notice that $\epsilon$ ​​​​​​ is independent of x and y.

As we know any function f is said to be uniformly continuous if

such that for all x and y in R, we have

0 < SE'O < PA

.

1 - y <Sf(2) - f(y)<E

that is epsilon and delta doesn't depend on x and y.

Thus given function f is a uniformly continuous function.

Other approch:

If a=0, then f is a constant function, therefore it is a uniformly continuous.

If a is nonzero then

brie f(n)= an+b. Now. derivative of fi f'(n) = a. XHEIR le et to 11'()] + [al. thus, f(n) is bounded. which means of is a Libechitz

As we know every Lipschitz function is uniformly continuous. Therefore f is a uniformly continuous function.