4
$\begingroup$

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that the set $A:=\{x\in\mathbb{R};f'(x)\not\in\{1,-1\}\}$ has measure $0$. Does this imply that $f'$ is constant?

Context: I was thinking about a question whose author probably meant "continuously differentiable curves" instead of just "differentiable curves", but I wanted to think about the general case. I don't know anything about functions with discontinuous derivative so I have no idea if the answer will be yes or no.

Edit: As Christian Remling mentions in the comments, here is another answer which implies that if $f'$ was not constant, then the measure of $A$ would be greater than $0$. Also, as mentioned by Mateusz Kwaśnicki below, a paper was devoted to proving this result.

Improve this question
$\endgroup$
5
  • 7
    $\begingroup$ The derivative of an everywhere differentiable real-valued function satisfies the intermediate value property (a theorem of Darboux). Hence if $f'$ takes 2 different values, say $f'(a)=y_a$ and $f'(b)=y_b$ with, say, $y_a<y_b$, then $f'([a,b])$ contains the interval $[y_a,y_b]$. I'm pretty sure that prevents $A$ from having 0 Lebesgue measure, except if $f'$ is constant (I may be wrong, though). $\endgroup$ Commented Dec 14, 2022 at 21:38
  • 2
    $\begingroup$ This is answered here: mathoverflow.net/questions/266377/… $\endgroup$ Commented Dec 14, 2022 at 22:05
  • $\begingroup$ Thanks to both! $\endgroup$ Commented Dec 14, 2022 at 22:11
  • $\begingroup$ To spell this out more explicitly (it is trivial, but perhaps mildly confusing for a moment): If $f'$ is not constant, let's say $f'(a)=1$, $f'(b)=y\not=1$, apply the linked answer with $y$ taking the role of $-1$. $\endgroup$ Commented Dec 14, 2022 at 22:12
  • 4
    $\begingroup$ You will like this paper :-) $\endgroup$ Commented Dec 14, 2022 at 22:39

0

Reset to default

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.