---
tag: P5AWRD2793
kind: concept
name: Singular function
msc: [26A30]
sources:
  - url: https://en.wikipedia.org/wiki/Singular_function
same_as:
  wikidata: Q2374166
  wikipedia: Singular function
---

## Summary

A singular function is a continuous, non-constant function on an interval whose derivative exists and equals zero almost everywhere.
Such a function cannot be recovered by integrating its derivative.
So it shows that the [[fundamental theorem of calculus|RX7JSHMP8Z]] needs [[absolute continuity|ZT6F4C2YHK]] in the Lebesgue setting.
The [[Cantor function|NPBYYK63FE]] and the [[Minkowski question-mark function|4D9HP5TS6W]] are the standard examples.
Every function of [[bounded variation|A8251BHCMP]] is a sum of three parts.
They are an absolutely continuous function, a jump function and a continuous function with derivative $0$ almost everywhere.

## Explanations

### L1 human

Picture a staircase with infinitely many tiny steps, where the flat parts of the stairs fill almost all of the interval.
The function climbs from one value to another, yet at almost every point its slope is zero.
All the growth happens on a set of length zero.
The Cantor function is such a staircase: it is flat on every interval removed in the construction of the Cantor set, and it still climbs from $0$ to $1$.