Suppose $f$ is uniformly continuous on $[a,b]$. Does it mean $f$ is also uniformly continuous on $(a,b)$?

Question

I am just curious if we can use the definition $\forall x, y \in [a,b]$ and $\exists \epsilon > 0$ s.t $|f(x) - f(y)| < \delta$ if $|x-y|<\epsilon$ . Then we just exclude the endpoints. Would that work? or is the statement too obvious and I am just over thinking? Thanks

Answer 1

Yes, that would be correct. More generally: if $f$ is a uniformly continuous function from $A(\subset\mathbb{R})$ into $\mathbb R$ and if $B\subset A$, then $f$ is also uniformly continuous on $B$ (by the same argument).