Clarification on a statement of a theorem

Question

If $f(x)$ be continuous over $(0,1]$ then $f$ is uniformly continuous if and only if $\lim_{x \to 0^+} f(x)$ exists.

I have a question about this statement, I am unclear on whether if it is saying that $f$ is uniformly continuous only over $(0,1]$ or is it everywhere, where $f$ is defined? Thanks!!!

Answer 1

Only over $(0,1]$; since you don’t know anything about the function elsewhere, that’s the only place you know it’s uniformly continuous.