Singular function

Builds on Continuous function Derivative

Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a function. The function $f$ is singular if:

$f$ is continuous on $[a, b]$ ;
there is a set $N \subseteq [a, b]$ of Lebesgue measure $0$ such that for every $x \in [a, b] \setminus N$ the derivative $f'(x)$ exists and $f'(x) = 0$ ;
$f$ is not constant on $[a, b]$ .

Remarks

Example: the Cantor function $c$ on $[0, 1]$ is singular. It is continuous, it is constant on each open interval removed in the construction of the Cantor set, so $c' = 0$ off the Cantor set, which has measure $0$ , and $c(0) = 0 \ne 1 = c(1)$ .

Non-example: the step function equal to $0$ on $[0, \frac12)$ and $1$ on $[\frac12, 1]$ satisfies clauses 2 and 3 but fails clause 1.

Conventions vary: some authors drop continuity, and some allow constants. A singular function is never absolutely continuous, since an absolutely continuous function with derivative $0$ almost everywhere is constant.

Reading path

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Limit point of a set definition
Limit of a function definition
Continuity at a point definition
Derivative definition
Continuous function definition
Singular function definition

Tag	`2YYCNFPX1F`
URI	`m2://2YYCNFPX1F`
Kind	definition
Writing	W0 stub
MSC	26A30
Defines	Singular function
Uses	Continuous function, Derivative
Changed	2 November 2026

Singular function

Definition

Remarks

Reading path

Further reading

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