Is every discontinuities function is integrable?

Not every function with discontinuities is integrable. A function is considered integrable (in the Riemann sense) if the set of its discontinuities has measure zero. Functions with too many or too "wild" discontinuities, such as the Dirichlet function (which is 1 for rational numbers and 0 for Irrational Numbers), are not Riemann integrable. However, they may still be Lebesgue integrable under certain conditions.