My lecturer made the following claim, which I have been trying hard to prove:
Suppose $f:\mathbb R\to \mathbb R$ is strictly increasing, continuous and $f$ is twice differentiable at some point $x$ satisfying $f'(x)\neq 0$. Then $f$ is continuously differentiable in a neighbourhood of $x$.
Presumably this depends crucially on the strictly increasing hypothesis, but I can't manage to make use of this fact!