Prove that $f$ is uniformly continuous, when $f''$ is bounded.

Question

Let $f$ be a twice differentiable function on $(0,1)$. It is given that for all $x\in(0,1)$, $|f''(x)|\leq M$; where $M$ is a non-negative real number. Prove that $f$ is uniformly continuous on $(0,1)$

Answer 1

Since $f''$ is bounded, it follows that $f'$ is uniformly continuous by the mean value theorem. So we can extend $f'$ to the boundary and make it a continuous function on $[0,1]$. But now this is a continuous function on a compact set, and so is also bounded. So we can descend again to $f$.