The question:

Given , prove:

a) isn't uniformly continuous functions on

b) is uniformly continuous functions on

My solution(check me please):

a) First I say,

,

now,

becomes to be function with one variable- :

.

Now I pick two points :

,

We can see that:

But:

Therefor for every , so that for all will be held:

b)

is closed and bounded, the function is continuous function in , hence from Cantor's theorem we can deduce that is uniformly continuous function.