1
$\begingroup$

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 .

New contributor
user165331 is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
$\endgroup$
  • 3
    $\begingroup$ Can you give some context for your question? This sounds a bit like homework from the way it is asked. $\endgroup$ – Anthony Quas Sep 14 at 15:45
  • $\begingroup$ 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. $\endgroup$ – Iosif Pinelis Sep 14 at 16:01
  • 1
    $\begingroup$ @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 $\endgroup$ – user165331 Sep 14 at 16:12
  • $\begingroup$ 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. $\endgroup$ – Iosif Pinelis Sep 14 at 17:42
  • 1
    $\begingroup$ @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 . $\endgroup$ – user165331 Sep 14 at 18:14

Your Answer

user165331 is a new contributor. Be nice, and check out our Code of Conduct.

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.