A functional equation

Question

Let $u$ be a mapping from $(0,1)$ to its self. I m interested in a class of functions $u$ for which the following equation admits a solution $f: (0,1) \to R$ which is derivable in $(0,1)$ :

$$ f\circ u= f +1 $$

Answer 1

If you take for example

$u\;:\;x\mapsto x$

then, there is no solution.