For arbitrary real function $f(x)$ from $[0,\infty)$ to $[0,\infty)$ which is continuous, differentiable and increasing strictly define real function $g(x)$ to be inverse of $f(x)$ then is it possible to find all $f(x)$ such that $ g(x)=\frac{1}{2}f'(\sqrt{x})$ holds?
sure, $f(x) = x^2$ is one possibility but couldn't there be no more? Thus unique solution? Can i solve it by techniques of DEs?