Using Darboux sums, if $f$ is a piecewise continuous function in $[a,b]$, then
It is integrable in $[a,b]$.
Given $\epsilon>0$, there is $\delta>0$ such that for every $P$ partition:
$$||P|| < \delta\implies S\left(f,P\right) - I\left(f,P\right) < \epsilon$$
I already know how to prove the first proposition, my question is for the second one. Please and thank you. This can be proven adding a point $c$ to $[a,b]$.