# Dirichlet’s function

Dirichlet’s function $f:\mathbb{R}\to\mathbb{R}$ is defined as

 $f\left(x\right)=\left\{\begin{array}[]{ll}\frac{1}{q}&\textrm{if }x=\frac{p}{q% }\textrm{ is a rational number in lowest terms,}\\ 0&\textrm{if }x\textrm{ is an irrational number.}\end{array}\right.$

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

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

 $f\left(x\right)=\left\{\begin{array}[]{ll}1&\textrm{if }x\textrm{ is an % rational number.}\\ 0&\textrm{if }x\textrm{ is an irrational number.}\end{array}\right.$

This nowhere-continuous function has the surprising expression

 $\displaystyle f(x)=\lim_{m\to\infty}\lim_{n\to\infty}\cos^{2n}(m!\pi x).$

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 DirichletsFunction 2013-03-22 13:11:14 2013-03-22 13:11:14 mathcam (2727) mathcam (2727) 9 mathcam (2727) Definition msc 26A15 FunctionContinuousAtOnlyOnePoint APathologicalFunctionOfRiemann