# 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} 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 ExampleOfANonRiemannIntegrableFunction 2013-03-22 15:03:28 2013-03-22 15:03:28 paolini (1187) paolini (1187) 4 paolini (1187) Example msc 28-XX msc 26-XX