---
tag: 04YKAYDQPW
kind: statement
role: theorem
name: Monotone functions are Riemann integrable
msc: [26A42]
uses: [017DH8WXTE, 0TR9SR9FTB]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann_integral
    locator: Integrability
---

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be monotone.
Then $f$ is Riemann integrable on $[a, b]$.

## Remarks

Assume $f$ increasing, and take the partition into $n$ equal pieces. Then $M_i = f(x_i)$ and $m_i = f(x_{i-1})$, so the sum telescopes:
$$U(f, P) - L(f, P) = \frac{b - a}{n} (f(b) - f(a)).$$
This tends to $0$, so [[Riemann's criterion|017DH8WXTE]] applies.
A monotone function can have infinitely many discontinuities, and it is still integrable.

## Explanations

### L1 human

For an increasing function, the gaps between upper and lower rectangles stack up to one column of total height $f(b) - f(a)$.
Its width shrinks to $0$ as the partition gets finer.