Suppose that we have a real valued function $f(x)$ and define the function $g(x)$ such that $g(g(x))=f(x)$. What is the derivative of $g(x)$? And there is some rule that express this derivative in terms of the derivative of $f(x)$?
It seems a simple question but, searching on the web, I found references to the functional square root of a function (which refer to the Shroeder's functional equation for generalizations), but noting about the derivation of such function.
Someone knows some useful reference to this topic and, more in general, to fractional iteration of functions?
Without additional assumptions about the function $g(x),$ I suspect little can be said. For instance, let $g(x)$ be the Dirichlet function that has the value $1$ when $x$ is rational and has the value $0$ when $x$ is irrational. Then $g(g(x)) = 1$ for all $x,$ and thus is about as nice a function that one could hope for, but $g(x)$ is not differentiable at any point. Indeed, $g(x)$ is not continuous at any point and not Riemann integrable on any interval. Also, if $E$ is a set that doesn't contain $0$ or $1$ and we let $g(x)$ be the characteristic function (or indicator function) of $E,$ then $g(g(x)) = 0$ for all $x.$ By choosing $E$ appropriately, the function $g(x)$ can simultaneously be nowhere continuous, not Borel measurable, not Lebesgue measurable, etc.