---
tag: AD0G065GK8
kind: example
name: Pompeiu derivative
msc: [26A24, 26A27]
illustrates: [0V1EJCT8G6, 7BJEH1GCTE]
uses: [0V1EJCT8G6, 0BSJSX7ME2, 011EJQWWJS]
sources:
  - url: https://en.wikipedia.org/wiki/Pompeiu_derivative
---

## Example

Let $h$ be a function that is differentiable at every point.
Its [[derivative|0V1EJCT8G6]] $h'$ is a Pompeiu derivative if $h'$ vanishes on a [[dense set|0BSJSX7ME2]] but is not identically zero.
Here is a construction.
Let $q_1, q_2, \dots$ list the rational numbers in $[0, 1]$, and write $u^{1/3}$ for the real cube root.
For $x \in [0, 1]$ set
$$g(x) = \sum_{n=1}^\infty 2^{-n} (x - q_n)^{1/3}.$$
The series converges uniformly, and each term is increasing, so $g$ is continuous and [[strictly increasing|011EJQWWJS]].

Each difference quotient of $u \mapsto u^{1/3}$ at a point $u \ne 0$ is positive and at most $\frac{4}{3} \abs{u}^{-2/3}$, since $a^2 + ab + b^2 \ge \frac{3}{4} a^2$ for real $a, b$.
Use this bound with dominated convergence where the series below is finite, and Fatou's lemma where it is infinite.
It follows that $g$ has at each $x$ a derivative in $(0, \infty]$, namely
$$g'(x) = \sum_{n=1}^\infty \frac{2^{-n}}{3} \abs{x - q_n}^{-2/3},$$
with $g'(q_n) = \infty$ for every $n$ (one-sided at the end points).
Since $\abs{x - q_n} \le 1$, every term is at least $2^{-n}/3$, so $g'(x) \ge 1/3$.

Let $h = g^{-1}$ on the interval $[g(0), g(1)]$.
By the rule for the derivative of an inverse function, $h$ is differentiable everywhere, with $h'(y) = 1/g'(h(y))$, read as $0$ when $g'(h(y)) = \infty$.
So $0 \le h' \le 3$.
The derivative $h'$ vanishes at every point $g(q_n)$, and these points are dense in $[g(0), g(1)]$ because $g$ is a homeomorphism.
But $h'$ is not identically zero, since $h$ is strictly increasing and a function with zero derivative is constant.
So $h'$ is a Pompeiu derivative.

## Explanations

### L1 human

One might think that a derivative which is zero on a dense set of points must be zero everywhere.
This example shows that this is false.
The function $g$ has vertical tangents at all rational points, so its inverse $h$ has horizontal tangents at a dense set of points, and yet $h$ keeps increasing.
Like every derivative, $h'$ still has the intermediate value property, so it is a [[Darboux function|7BJEH1GCTE]].