Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$.

By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \in [a,b]: |f'(x)| < 1\}$ is uncountable. But how small can it be? Or to be more formal:

Can $\{x \in [a,b]: |f'(x)| < 1\}$ have measure zero?

I guess not, because I have never heard of such a counterexample. But I don't see how to prove it.