$$f(x) = \left\{ \begin{array}{11} x & \mbox{when $x$ is rational} \\ (-x) & \mbox{when $x$ is irrational} \end{array} \right. $$
Prove that $f(x)$ is not integrable over $[a, b]$
I thought along the line of finding an epsilon $\epsilon$ such that $ U(P, f) - L(P, f) >\epsilon$ but cannot proceed. Need help Thanks in advance
hint
$$M_i=\sup_{[x_i,x_{i+1}]}f=-\inf_{[x_i,x_{i+1}]}=m_i$$
where $(x_i)$ is a partage $ P $ of $[a,b]$ .
$$U(P,f)=\sum_{i=0}^{n-1}M_i(x_{i+1}-x_i)=-L(P,f)=-\sum_{i=0}^{n-1}m_i(x_{i+1}-x_i)$$