example of a non Riemann integrable function

Let $[a,b]$ be any closed interval and consider the Dirichlet’s function $f\colon[a,b]\to\mathbb{R}$

f(x)=\begin{cases}1&\text{if $x$ is rational}\\ 0&\text{otherwise}.\end{cases}

Then $f$ is not Riemann integrable. In fact given any interval $[x_{1},x_{2}]\subset[a,b]$ with $x_{1}<x_{2}$ one has

\sup_{[x_{1},x_{2}]}f(x)=1,\qquad\inf_{[x_{1},x_{2}]}f(x)=0

because every interval contains both rational and irrational points. So all upper Riemann sums are equal to $1$ and all lower Riemann sums are equal to $0$ .

Title	example of a non Riemann integrable function
Canonical name	ExampleOfANonRiemannIntegrableFunction
Date of creation	2013-03-22 15:03:28
Last modified on	2013-03-22 15:03:28
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	4
Author	paolini (1187)
Entry type	Example
Classification	msc 28-XX
Classification	msc 26-XX