Definition:Definite Integral/Darboux

Definition
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ be a real function. Let $f$ be bounded on $\left[{a \,.\,.\, b}\right]$.

Suppose that:
 * $\displaystyle \underline{\int_a^b} f \left({x}\right) \ \mathrm dx = \overline{\int_a^b} f \left({x}\right) \ \mathrm dx$

where $\displaystyle \underline{\int_a^b}$ and $\displaystyle \overline{\int_a^b}$ denote the lower integral and upper integral, respectively.

Then the definite (Riemann) integral of $f$ over $\left[{a \,.\,.\, b}\right]$ is defined as:
 * $\displaystyle \int_a^b f \left({x}\right) \ \mathrm d x = \underline{\int_a^b} f \left({x}\right) \ \mathrm dx = \overline{\int_a^b} f \left({x}\right) \ \mathrm dx$

$f$ is formally defined as (properly) integrable over $\left[{a \,.\,.\, b}\right]$ in the sense of Riemann, or (properly) Riemann integrable over $\left[{a \,.\,.\, b}\right]$.

More usually (and informally), we say:
 * $f$ is (Riemann) integrable over $\left[{a \,.\,.\, b}\right]$.

Also see

 * Equivalence of Definitions of Riemann Integral