Proposition 6 (theorem of the mean)

Tag 087O, permanent. Cite it as [Bourbaki, Tag 087O].

Functions of a Real Variable, chapter II, § 1. PRIMITIVES AND INTEGRALS

Let $f$ be a real regulated function on a compact interval $l = [a, b]$ ; if $J$ is the set of points of I where $f$ is continuous, and $m = \inf_{x \in J} f(x)$ , $M = \sup_{x \in J} f(x)$ , then

m < \frac{1}{b-a} \int_a^b f(t)\,dt < M

(13)

except when $f$ is constant on J, in which case the three members of (13) are equal.

In other words, the mean of the regulated function $f$ in I lies between the bounds of $f$ over the subset of I where $f$ is continuous.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.