Cauchy Criterion

Theorem

A bounded function $f:[a, b] \rightarrow R$ is Riemann integrable iff for every $\epsilon > 0$ there exists a partition $P_\epsilon$ of $[a,b]$, such that:

$$U(f, Pϵ) − L(f, Pϵ) ≤ \epsilon$$

Proof Hint

• To prove $\implies$: consider $L(f)-\frac{\epsilon}{2} \lt L(f;P)$ and $U(f;P) \lt U(f)+\frac{\epsilon}{2}$
• To prove $\impliedby$: consider $L(f;P)<L(f)$ and $U(f)<U(f;P)$

Note

$f:[a,b]\rightarrow\mathbb{R}$ is integrable on $[a,b]$ when:

• The set of points of discontinuity of a bounded function $f$ is finite.
• The set of points of discontinuity of a bounded function $f$ is finite number of limit points. (may have infinite number of discontinuities)

In these cases, the discontinuities don’t affect the result of the integration.