**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 https://math.stackexchange.com/questions/3312572/questions-about-the-existence-of-a-function?noredirect=1#comment6815760_3312572.

Now, I want to attack the initial problem. Previous arguments do not work!

Do you have any suggestions for me?

I have already asked the question https://math.stackexchange.com/questions/3313126/existence-of-function-satisfying-ffx-x-almost-everywhere, but the subject will be closed for a reason that I do not understand

I think, we need other non-classical arguments