Pointwise and uniform convergence difference

Question

Why when expressing pointwise convergence we use a single set (e.g. $f(x)$ pointwise convergent on $[-1, 1]$) and when expressing uniform convergence we say something like "$f(x)$ uniformly convergent on all $[-a, a] \subset (-1, 1)$"?

Answer 1

I think you are confused about two things.

First, a function can't be uniformly convergent - it can be uniformly continuous. A sequence of functions can be uniformly convergent.

I'll assume you are thinking about a sequence.

Then the statement

"$f_n(x)$ is uniformly convergent on all $[−a,a]⊂(−1,1)$"

is not the way we talk about uniform convergence in general. In this particular case the statement says that although the sequence $f_n$ is not uniformly convergent on the open interval, it is uniformly convergent on every closed subinterval.