Tuesday, December 26, 2017

real analysis - Can it be that $f$ and $g$ are everywhere continuous but nowhere differentiable but that $f circ g$ is differentiable?

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.

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