DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passaudited 2026-08-11

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Improper integrals at a finite singular endpoint

Statement

Suppose $f$ is Riemann integrable on every $[c, b]$ with $a < c < b$ . Its improper integral at the left endpoint is $\int_{a}^{b} f (x) d x : = \lim_{c ↓ a} \int_{c}^{b} f (x) d x,$ provided the one-sided limit is a finite real number.

If instead $f$ is integrable on every $[a, c]$ with $a < c < b$ , define $\int_{a}^{b} f (x) d x : = \lim_{c ↑ b} \int_{a}^{c} f (x) d x .$ Convergence always means existence of the displayed finite limit. The value assigned to $f$ at the singular endpoint, if any, is irrelevant because changing one endpoint value does not change any proper truncation integral.

Depends on

Dependency tree · 2 levels

Sources

William F. Trench, Introduction to Real Analysis, Section 3.4