Dirichlet Function¶

Definition¶

The Dirichlet function is the indicator function of the rational numbers, defined by

f(x)=\indicator_{\mathbb{Q}}(x)=\begin{cases} \displaystyle \,1,& \text{if }x\in\mathbb{Q},\\[2pt] \displaystyle \,0,& \text{if }x\notin\mathbb{Q}. \end{cases}

If the input $x$ is a rational number, the output of the Dirichlet function is $1\text{;}$ otherwise, if the input $x$ is an irrational number, the output of the Dirichlet function is $0\text{.}$

Interpretation¶

The Dirichlet function provides a fundamental example of extreme irregularity. It sorts real numbers into rational and irrational ones, giving the value $1$ to the first kind and $0$ to the second.

The irregularity comes from how the two kinds are mixed: every interval $(a,b)$ with $a<b$ $b$ , however short, contains both rational and irrational numbers. So close to any point $x$ , the function takes both values $0$ and $1$ , and its values never settle down near $x$ . A sketch of the graph as two horizontal lines $y=0$ and $y=1$ is misleading: each of these "lines" has gaps between any two of its points.

The function is the indicator function of $\mathbb{Q}$ . Later it serves as the standard example of a function that is continuous at no point, and of a bounded function that is not Riemann integrable.

Key Results¶

The Dirichlet function is discontinuous at every point.
It is not Riemann integrable on any nondegenerate bounded interval.

Examples¶

On any nondegenerate interval, rational and irrational inputs are densely interwoven, preventing continuity.
Any neighbourhood contains points where the function value jumps between $0$ and $1\text{.}$

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 4

Keywords: nowhere continuous function Symbols: element of, rational numbers

Last updated: 20 October 2026 03:02