If $f : [a,b] \rightarrow \mathbb{R}$ is uniformly continuous, then $f : (a,b) \rightarrow \mathbb{R}$ is uniformly continuous

Question

If $f : [a,b] \rightarrow \mathbb{R}$ is uniformly continuous, then $f : (a,b) \rightarrow \mathbb{R}$ is uniformly continuous

This solution is for the reverse statement. Does the example in this still stand then as a counterexample?

Answer 1

Hint

$f$ uniformly continuous at $(a,b)$

$\implies$

$\lim_{x\to a^+}f(x)=L^+ $ and

$\lim_{x\to b^-}f(x)=L^-$ exist.

using sequential caracterisation of the limit and Cauchy criteria.

so, we can extend $f$ to $g$ defined on $[a,b]$ by

$g(a)=L^+$ and $g(b)=L^-$.

we prove that $g$ is uniformly continuous at $[a,b]$.