So, I was just asking myself can something like this happen? I was thinking about some everywhere continuous but nowhere differentiable functions $f$ and $g$ and the natural question arose on can the composition $f \circ g$ be differentiable, in other words, can the operation of composition somehow "smoothen" the irregularities of $f$ and $g$ which make them non-differentiable in such a way that composition becomes differentiable?
So here is the question again:
Suppose that $f$ and $g$ are everywhere continuous but nowhere differentiable functions. Can $f \circ g$ be differentiable?
If such an example exists it would be interesting because the rule $(f(g(x))'=f'(g(x)) \cdot g'(x)$ would not hold, and not only that it would not hold, it would not make any sense because $f$ and $g$ are not differentiable.