Besides Dirichlet functions, which functions are Riemann non integrable and Lebesgue integrable,
{a (not 0) x = rational number
f(x)=
{0 x = irrational number
As long as the function is not continuous or finite continuous
{a (not 0) x = rational number
f(x)=
{0 x = irrational number
As long as the function is not continuous or finite continuous