L01 · Riemann integral — setup

Upper & lower sums

For a bounded $f:[a,b]\to\mathbb{R}$ and partition $P$ , $U(f,P)=\sum M_i\Delta x_i$ and $L(f,P)=\sum m_i\Delta x_i$ .

Prove that if $f$ is continuous on $[a,b]$ , then $f$ is Riemann integrable.

Solution · P1

Use uniform continuity on the compact interval to make $U(f,P)-L(f,P)$ arbitrarily small.