Real Analysis question continuity question where a proof is needed.

Question

Suppose that $f(x)$ is a continuous function on $(a,b)$ such that: $$\lim_{x\to a+} f(x)=A, \lim_{x\to b-} f(x)=B$$ Prove that $f(x)$ must be uniformly continuous

Answer 1

Hint:

This implies $f$ can continuously be extended to $[a,b]$

Answer 2

Define $g:[a,b] \to \mathbb R$ by

$g(x):=f(x)$, if $x \in (a,b)$, $g(a):=A$ and $g(b):=B$.

Since $\lim_{x\to a+} f(x)=A$ and $ \lim_{x\to b-} f(x)=B$, $g$ is continuous.

$[a,b]$ is compact, hence $g$ is uniformly continuous.

Therefor $f$ is uniformly continuous.