Given a continuous function $f(x)$ when does there exist a non-constant continuous function $g(x)$ such that $f(g(x))$ is differentiable what about $g(f(x))$?

Does there exist any examples of $f(x)$ such that no such $g(x)$ exist for either of the transformations ?

*In this paper*-

Bruckner, A. M. “Creating Differentiability and Destroying Derivatives.” The American Mathematical Monthly, vol. 85, no. 7, 1978, pp. 554–562. JSTOR, www.jstor.org/stable/2320863.

A characterisation for $g(x)$ being a homeomorphism is given .

It is easy to construct examples for where no such $g(x)$ exists if it is restricted to a homeomorphism using nowhere differentiable functions .

existsan anti-derivative function), the mentioned characterization contains the non-constructive condition of theexistenceof a (gauge) function. $\endgroup$ – Iosif Pinelis Sep 14 at 17:42