Chapter 5.2, Problem 88E

Integrating piecewise continuous functions Suppose f is continuous on the intervals [a, p] and [p, b], where a < p < b, with a finite jump at p. Form a uniform partition on the interval [a, p] with n grid points and another uniform partition on the interval [p, b] with m grid points, where p is a grid point of both partitions. Write a Riemann sum for ${\displaystyle {\int }_a^{b}f(x)dx}$ and separate it into two pieces for [a, p] and [p, b]. Explain why ${\displaystyle {\int }_a^{b}f(x)dx={\displaystyle {\int }_a^{p}f(x)dx+{\displaystyle {\int }_p^{b}f(x)dx}}}$.