Contraction Theorem & Measure Zero

Question

It is well known that if we have an interval map $f:I\rightarrow I$ with $f'(x)<1$ for all $x\in I$, then $f$ is a contraction.

I want to understand the "limits" of this lemma. Say on a set of measure zero $f'(x)=1$, will the result still hold? I feel like it won't hold even if $f'(x)=1$ at a single point, but I can't formalize the argument.

Answer 1

If $f$ is absolutely continuous, $f(a) - f(b) = \int_a^b f'(x)\; dx$. So if $|f'| < 1$ almost everywhere and $a < b$, $$|f(a)-f(b)| \le \int_a^b |f'(x)|\; dx < b-a$$