Show that $f$ is not Riemann integrable on $[0,1]$ If $x$ is any rational number, $f(x)=0$. If $x$ is any irrational number, $f(x)=1$. I know that $f(x)$ oscillate between $0$ and $1$ on $[0.1]$. But I have not idea why it isn't integrable on $[0.1]$.

I assume that you mean that this function is not Riemann Integrable, as this function is actually Lebesgue Integrable. A function is defined as Riemann integrable if the upper sums and the lower sums of arbitrary partitions get arbitrarily close.

Let $0=x_0
Hope this helps!