Riemann Integrability Criterion

The Riemann Integrability Criterion characterizes which functions are Riemann integrable. A bounded function is Riemann integrable if and only if it is continuous almost everywhere (the set of discontinuities has measure zero). For practical purposes, continuous functions and bounded functions with finitely many discontinuities are integrable.

Statement

Theorem6.1Riemann's Criterion

A bounded function $f : [a, b] \to \mathbb{R}$ is Riemann integrable if and only if for every $\epsilon > 0$ , there exists a partition $P$ such that

$U(f, P) - L(f, P) < \epsilon,$

where $U(f, P)$ and $L(f, P)$ are the upper and lower sums.

Theorem6.2Continuous functions are integrable

If $f : [a, b] \to \mathbb{R}$ is continuous, then $f$ is Riemann integrable.

ProofSketch

By uniform continuity (via Heine-Cantor), for any $\epsilon > 0$ , there exists $\delta > 0$ such that $|f(x) - f(y)| < \epsilon/(b-a)$ whenever $|x - y| < \delta$ . Choose a partition $P$ with mesh $< \delta$ . Then on each subinterval, $M_i - m_i < \epsilon/(b-a)$ , so

$U(f, P) - L(f, P) = \sum_{i=1}^n (M_i - m_i)(x_i - x_{i-1}) < \frac{\epsilon}{b-a} \sum_{i=1}^n (x_i - x_{i-1}) = \epsilon.$

Thus $f$ is Riemann integrable.

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Theorem on discontinuities

Theorem6.3Integrability and discontinuities

A bounded function $f : [a, b] \to \mathbb{R}$ is Riemann integrable if and only if the set of discontinuities of $f$ has Lebesgue measure zero.

RemarkPractical consequence

Functions with finitely many discontinuities are Riemann integrable (since finite sets have measure zero). Monotone functions are integrable (they have at most countably many discontinuities, which has measure zero).

ExampleStep functions are integrable

A step function (piecewise constant with finitely many jumps) has finitely many discontinuities, hence is Riemann integrable.

ExampleMonotone functions are integrable

If $f$ is monotone on $[a, b]$ , then $f$ has at most countably many discontinuities (all jumps), hence is Riemann integrable.

Summary

The integrability criterion characterizes Riemann integrable functions:

$f$ is integrable $\Leftrightarrow$ $U(f, P) - L(f, P)$ can be made arbitrarily small.
Continuous functions are integrable (by uniform continuity).
Functions with finitely many discontinuities are integrable.
Functions with measure-zero discontinuity sets are integrable.

See Fundamental Theorem for evaluating integrals.