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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?

Paul
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  • @Kavi Rama Murthy Can u explain me why u voted against this question? – Paul Aug 04 '19 at 15:03
  • @Jack D'Aurizio
    in the above link, I explained this case thank you for saving my question.     – Paul Aug 04 '19 at 18:51
  • The question is not yet solved in https://mathoverflow.net/questions/337607/existence-of-function-satisfying-ffx-x-almost-everywhere – Paul Aug 05 '19 at 21:57
  • @Jack D'Aurizio
    the link https://math.stackexchange.com/questions/3312572/questions-about-the-existence-of-a-function?noredirect=1#comment6815760_3312572 does not answer the question you asked me?     – Paul Aug 06 '19 at 18:18
  • Convert your equation into a differential equation and seek out for a solution – ibnAbu Aug 08 '19 at 03:07
  • @ ibnAbu can explain your approach ?! – Paul Aug 08 '19 at 14:28
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    Why this question is different from?$$f'(x)=f^{-1}(x)$$ which is solvable, see video – Joako Sep 12 '24 at 18:16

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