# Transforming a continuous function into a differentiable function

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 .

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• Can you give some context for your question? This sounds a bit like homework from the way it is asked. – Anthony Quas Sep 14 at 15:45
• In what terms do you want your characterizations to be expressed? For any condition $C$, the tautological characterization is always available: $C$ holds iff $C$ holds. So, without specifying the terms, your problems lose meaning. – Iosif Pinelis Sep 14 at 16:01
• @IosifPinelis If one wants to characterise the derivative simply saying F is a derivative if there exists G such that G’ = F is enough but this does not reveal anything new about derivatives . But Derivatives have interesting properties such as they are baire 1 and they can’t be discontinuous everywhere etc. The tautology can’t be accepted as it does not reveal much (except the definition) about the structure of derivatives – user165331 Sep 14 at 16:12
• Proposition 1 in the paper projecteuclid.org/download/pdf_1/euclid.rae/1337001388 shows that moving a quantifier in the definition of the KH integrability provides a characterization of derivatives. However, I don't think this characterization is very useful: just as the definition of a derivative function (saying there exists an anti-derivative function), the mentioned characterization contains the non-constructive condition of the existence of a (gauge) function. – Iosif Pinelis Sep 14 at 17:42
• @IosifPinelis 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. Accessed 14 Sept. 2020. A characterisation for g(x) being a homeomorphism is given . – user165331 Sep 14 at 18:14