Dirichlet’s function

Dirichlet’s function f: is defined as

f(x)={1qif x=pq is a rational number in lowest terms,0if x is an irrational number.

This function has the property that it is continuous at every irrational number and discontinuousMathworldPlanetmath at every rational one.

Another function that often goes by the same name is the function

f(x)={1if x is an rational number.0if x is an irrational number.

This nowhere-continuous function has the surprising expression


This is often given as the (amazing!) example of a sequence of everywhere-continuous functions whose limit function is nowhere continuous.

Title Dirichlet’s function
Canonical name DirichletsFunction
Date of creation 2013-03-22 13:11:14
Last modified on 2013-03-22 13:11:14
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Definition
Classification msc 26A15
Related topic FunctionContinuousAtOnlyOnePoint
Related topic APathologicalFunctionOfRiemann