1

There are a number of continuous but nowhere-differentiable functions, none of which (to my knowledge) are bijections. I am wondering if continuous bijection $f:\mathbb{R} \to \mathbb{R}$ is necessarily everywhere-differentiable. I would think so, but am not sure how to formally approach this problem and have not found a similar question on MSE.

Descartes Before the Horse
  • 7,038
  • 1
    Related: https://math.stackexchange.com/questions/2853639/a-continuous-nowhere-differentiable-but-invertible-function (any such continuous bijection must be * almost everywhere* differentiable) – Clement C. Dec 04 '19 at 23:42
  • 1
    Not exactly the same -- you ask for everywhere differentiable (which is not necessarily true (in a strong sense), see note at the bottom of p.3 here: https://faculty.etsu.edu/gardnerr/5210/notes/6-2.pdf) – Clement C. Dec 04 '19 at 23:44
  • I voted to close this question but did not delete it because I was not able to find the original question, so anyone wondering the same thing who might not find the original as well could find this one. – Descartes Before the Horse Dec 04 '19 at 23:45

1 Answers1

4

If $f(x)$ is $x$ when $x\ge0$ and $2x$ when $x<0,$ $f$ is certainly a continuous bijection, but not differentiable at $x=0.$

Kenta S
  • 16,151
  • 15
  • 26
  • 53