Suppose that $f\colon (a, b)\to \mathbb R$ is a continuous function and there exists a point $p\in (a, b)$ such that $f'$ exists and is bounded on $(a, b)\setminus \{p\}$. Prove that $f$ is uniformly continuous on $(a, b)$
I am a little confused about the question. Whether $f'(p)$ exists or not in the description : "$f'$ exists and is bounded on $(a, b)\setminus \{p\}$". If $f'(p)$ exists, then I have proved it. But if $f'(p)$ doesn't exist?