Riemann integrability of monotone functions

Monotone functions are Riemann integrable: Let $a<b$. If $f:[a,b]\to\mathbb{R}$ is monotone , then $f$ is a Riemann integrable function on $[a,b]$.

This theorem is a cornerstone of the Riemann theory, providing integrability for many non-continuous examples; it complements criteria based on discontinuities and the oscillation criterion .