Corollary

Tag 0847, permanent. Cite it as [Bourbaki, Tag 0847].

Functions of a Real Variable, chapter I, § 2. THE MEAN VALUE THEOREM

Let $f$ be a finite continuous function on $[a,\ b]$ (where $a < b$ ) and have a right derivative (finite or not) at all points of the relative complement B in $[a,\ b[$ of a countable subset of this interval. If $m$ and $M$ are the greatest lower and least upper bounds of $f'_d$ on B then one has

m(b - a) < f(b) - f(a) < M(b - a)

if $f$ is not an affine linear function; if $f$ is affine linear one has

m = M = \frac{f(b) - f(a)}{b - a}.

The inequalities (5) are consequences of (4) when $m$ and $M$ are finite; the case when one or the other of these numbers is infinite is trivial.

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

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