I wasn't sure whether you knew measure theory; then I indeed meant "Lebesgue measurable".

Otherwise, Riemann integration suffices for piecewise continuous density functions (and a few others), like the usual ones.

Measurability is actually not defined with respect to a measure but to a $\displaystyle \sigma$-field ("une tribu"). I shouldn't have said "Lebesgue measurable", it was a slight mistake... (Or I could pretend I was talking about the Lebesgue $\displaystyle \sigma$-field)

The Lebesgue measure is defined on the Borel $\displaystyle \sigma$-field ("la tribu des boréliens"). Continuous functions are measurable with respect to this $\displaystyle \sigma$-field, but there are plenty of other measurable functions. So many that it is somewhat difficult to find a non-measurable one! In fact it is not possible to build non-measurable functions (with respect to the borelian $\displaystyle \sigma$-field) without using the axiom of choice. In practice, any function is measurable, except if it was meant to be a counter-example to this property

The counting measure is defined on the discrete $\displaystyle \sigma$-field, for which any function is measurable.