My question is related to this post.
What I don't quite get is why the OP's approach doesn't work. He said that not every Lipschitz function f on $[0,1]$ with Lipschitz constant less than 1 arises by considering $f(x)=f(0)+\int_0^xF(t)dt$ with $F$ being bounded by 1. I don't understand why not? because by Rademacher theorem, every $f\in Lip([0,1])$ can be written in this form or am I mistaken something?
