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:
 * $\displaystyle \underline {\int_a^b} f \left({x}\right) \rd x = \overline {\int_a^b} f \left({x}\right) \rd x$

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

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

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

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

Also known as
Because of the Equivalence of Definitions of Riemann and Darboux Integrals, and because the concept of the Darboux integral is generally considered to be easier to grasp than the Riemann integral, it is a popular approach on an introductory course of integral calculus to teach the Darboux integral, but then to call it the Riemann integral. Technically it is not, but the difference is ultimately immaterial.

Also see

 * Equivalence of Definitions of Riemann and Darboux Integrals