Piecewise continuous function

A function is piecewise continuous on a finite interval if it is continuous between finitely many breakpoints and has finite one-sided limits at those points and at the endpoints. On $[0,\infty )$, require this on every finite interval. A Heaviside step function is an example; a function with an infinite blow-up at an interior point is not. This condition guarantees local integrability while allowing jumps in an input.