show that $f'$ exists a.e. or give a counterexample

Question

Let $f:[0,1]\rightarrow \mathbb{R}$ be a function such that $f(x)-f(y)<x-y$ for any $x,y\in [0,1]$ and $x>y$. Is it true that $f'$ exists a.e.?

Answer 1

$f(x)-x$ is a decreasing function so it is differentiable a.e.. Hence $f(x)=(f(x)-x)+x$ is also differentiable a.e..