It's correct. In a), I think you have to denote the new by because it's not the same function (in particular has two variables whereas has only one). You can denote instead of and instead of to show that those two numbers depend on .
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.