My project is to Study the existence of a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ differentiable almost everywhere satisfying $ f\circ f'(x)=x$ almost everywhere $x \in \mathbb{R}$
I began the study by supposing $f\in C ^ 1(\mathbb{R}) $, I have shown that f does not exist.
After, I found some difficulties when we assume only f differentiable on $\mathbb{R}$, I had an answer using Darboux's theorem Questions about the existence of a function.
Now, I want to attack the initial problem. Previous arguments do not work!
Do you have any suggestions for me?