Riemann integrability

There are several ways to characterize if it’s possible to take a Riemann integral.

With upper and lower sums

A function is integrable if the Riemann lower and upper sum are equal.

Using continuity

Riemann integrabile $⟺$ continous almost everywhere

One of the coolest ways to check this is using the continuity with limsup and liminf functions

i.e. the function is Riemann integrable iff $|\{x|h(x)\ne H(x)\}|=0$