I am not here referring to the devil staircase, but to the question mark function. This is a strictly increasing function from $\mathbb{Q}$ to $\mathbb{Q}$, with derivative always $0$. I have two questions:

  1. Is this a surprising result (in the same way that continuous functions nowhere differentiable were thought surprising when discovered)?

  2. What are the properties preventing this (obviously, we need a topological field to speak of derivative; are there exemples of connected topological fields with non constant functions having everywhere a null derivative?)

  • What is a "topological field" and why do we need it to speak of the derivative? – GH from MO Aug 28 '12 at 14:44
  • 2
    A topological field is a field with a topology such that the operations $x\mapsto -x$ $x,y\mapsto x+y$ $x,y\mapsto xy$ and $x\mapsto x^{-1}$ are continuous. One will need a topology in order to take any sort of limit operation such as a derivative. – 35093731895230467514051 Aug 28 '12 at 15:30
  • @Joseph: Thanks for the clarification! – GH from MO Aug 28 '12 at 16:07

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.