$f:(a,b] \rightarrow \mathbb{R}$ continuous. $f'(x)$ exists on $(a,b)$ and $\lim_{x \rightarrow a+}f'(x)$ exists. Prove that $f$ is uniformly continuous on $(a,b]$
My thoughts: If we can define $f(a)$ such that $f(x)$ is continuous on $[a,b]$ we are done. But we only know $\lim_{x \rightarrow a+}f'(x)$ exists. How can we define $f(a)$ out of $f'(a)$?
Since the limit $\lim_{x\to a^+}f'(x)$ exists, there is some $c\in(a,b]$ such that $f'|_{(a,c]}$ is bounded, which implies that $f|_{(a,c]}$ is uniformly continuous. And si $[c,b]$ is a closed and bounded interval and $f$ is coninuous, $f|-{[c,b]}$ is also uniformly continuous.
Therefore, $f$ is uniformly continuous.