So I know you can integrate some discontinuous functions when the function is discontinuous at a finite number of points. So you can integrate for example this function on a interval [-1,3]:
$$ f(x):=\begin{cases} x & x< \frac{1}{4} \\ \frac{1}{x} & x\geq \frac{1}{4} \end{cases} $$
But can you integrate this function on a interval the discontinuous point as an end point? Like [-1,1/4]?