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Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$.

By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \in [a,b]: |f'(x)| < 1\}$ is uncountable. But how small can it be? Or to be more formal:

Can $\{x \in [a,b]: |f'(x)| < 1\}$ have measure zero?

I guess not, because I have never heard of such a counterexample. But I don't see how to prove it.

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  • $\begingroup$ If $|f'(x)|\le 1$ for all $x\in [a,b]$ then the answer is no. For if the answer were yes, then $f'(x)=1$ a.e. and in particular $f'$ is Lebesgue-integrable. Thus (e.g., Theorem 7.21 of Rudin's Real and Complex Analysis) the fundamental theorem of calculus holds, so $f(x) - f(a) = \int_a^x f'(t)\,dt = x - a$, i.e., $f(x)$ is actually the linear function $x-a+f(a)$ and in particular $f'(b) = 1 \ne -1$. So I expect the answer to be no in general because I don't think it can help you to allow $f'(x)$ to exceed 1. $\endgroup$ – Timothy Chow Apr 4 '17 at 21:00
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Fact 1 (Goldowsky-Tonelli): Let $F:(a, b) \to \mathbb{R}$ be continuous and have finite derivative everywhere. Suppose $F' \geq 0$ almost everywhere. Then $F$ is monotonically increasing.

For a proof of this, see Saks, Theory of the integral, Chapter 6, page 206.

Suppose $X = \{x \in [a, b]: -1 < f'(x) < 1\}$ has zero measure. Let $Y = \{x \in [a, b]: f'(x) \leq -1\}$ and $Z = \{x \in [a, b]: f'(x) \geq 1\}$

Claim 1: Every point of $X$ is a limit point of $Y$ and a limit point of $Z$.

Proof: Suppose for example $x \in X$ is not a limit point of $Y$ - the other case is similar. Let $I$ be an open interval around $x$ disjoint with $Y$. Then at almost every $y \in I$, $f'(y) \geq 1$. Using Fact 1, it follows that the function $y \mapsto f(y) - y$ is monotonically increasing on $I$ and hence for every $y \in I$, $f'(y) \geq 1$ which is impossible as $x \in Y \cap X$.

Claim 2: The set of points of continuity of $f' \upharpoonright \overline{X}$ is dense in $\overline{X}$ (the closure of $X$).

Proof: Well known (using Baire category theorem).

Now let $I$ be any open interval around $x \in X$. Then the supremum of $f' \upharpoonright I$ is at least $1$ and its infimum is at most $-1$ by Claim 1. By Darboux theorem, it follows that $f' \upharpoonright I$ and hence also $f' \upharpoonright (I \cap X)$ takes every value in $(-1, 1)$. So $f'$ is everywhere discontinuous on $\overline{X}$ which contradicts Claim 2.

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    $\begingroup$ For lazy readers, this post is showing that the answer to the question (can the set where the absolute value of the derivative is strictly less than $1$ have measure $0$?) is 'no'. $\endgroup$ – LSpice Apr 4 '17 at 21:26
  • $\begingroup$ Great! It seems the result used here is actually older than Goldowsky and Tonelli papers. See de La Vallée Poussin's Cours d'analyse infinitésimale (1914) which is cited by Goldowsky. $\endgroup$ – Siméon Apr 7 '17 at 8:48

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