A functionis Riemann integrable if it is continuous and bounded on a closed interval. Of course, if a function is differentiable then it is continuous and hence Riemann integrable but there are many examples of functions which are bounded but not continuous on a closed interval but which are still Riemann integrable.

A simple example is shown above. The function has a discontinuity atbut is still Riemann integrable because this discontinuity occupies an interval of length zero. In fact if we construct a partition consisting of subpartitions onthenis Riemann integrable onandsince it is bounded and continuous on these subintervals and on the subintervalwe have

andso(1)

Letthenso Riemann integrability amounts to being Riemann integrable on the intervalsThis can be extended, so that a function with a finite numbers of discontinuities, each of length zero, is Riemann integrable. Another example is given by the function