0

Let Dn denote "n times differentiable" and let Cn denote "n times continuously differentiable".

I'm interested in learning about a counterexample I do not recall ever seeing: a C1 function

   f : ℝ → ℝ

that is D2 nowhere.

And what about a function g : ℝ → ℝ that is D2 everywhere but C2 nowhere?

Dan Asimov
  • 1,157
  • 6
    Just pick a continuous function which is nowhere differentiable, $f,$ and define $$F(x)=\int_0^x f(t),dt.$$ Then $F'(x)=f(x),$ so $F$ has your property. – Thomas Andrews Apr 09 '23 at 19:57
  • 2
    For the first one, take a continuous function $g$ that is nowhere differentiable and consider $f(x)=\int_0^x g(t),dt$. The second one is impossible, see here. – Sassatelli Giulio Apr 09 '23 at 19:57
  • 1
    Concerning the first question it suffices to find a continuous function that is nowhere differentiable. Then its antiderivative would satisfy the requirements. Concerning the second question you can do the same trick and reduce the problem of finding $D^1 $ function which is not $C^1$ at every point – Ryszard Szwarc Apr 09 '23 at 19:58
  • I guess that takes care of both questions! I've probably seen the first question answered before, but not the second one. – Dan Asimov Apr 09 '23 at 20:00

0 Answers0