Remark 3

The conclusion of prop. 2 is not guaranteed if one merely supposes that the set $A$ of "exceptional" points is nowhere dense in I, but not countable (cf. I, p. 37, exerc. 3).

Prop. 2 entails the following fundamental theorem (which appears to be more general):

THEOREM 1 (mean value theorem). *Let $f$ and $g$ be two finite continuous real-valued functions defined on a closed bounded interval $I = [a, b]$ and having a

$^4$ A function defined on an interval $I \subset \mathbf{R}$ is said to be right continuous at a point $x_0 \in I$ if its restriction to the interval $I \cap [x_0, +\infty[$ is continuous at the point $x_0$ relative to this interval; it comes to the same to say that the right limit of this function exists at this point and is equal to the value of the function at this point.

right derivative (finite or not) at all points of the relative complement in $[a,\ b[$ of a countable subset of this interval. Suppose further that $f'_d(x)$ and $g'_r(x)$ are not simultaneously infinite except at the points of a countable subset of I and that there are finite numbers $m$, $M$ such that

except at the points of a countable subset of I (replacing $Mg'_r(x)$ (resp. $mg'_r(x)$) by 0 if $M = 0$ (resp. $m = 0$) and $g'_r(x) = \pm \infty$. Under these conditions one has

except when one has $f(x) = Mg(x) + k$, or $f(x) = mg(x) + k$ ($k$ constant) for all $x \in I$.

It suffices to apply prop. 2 to the functions $Mg - f$ and $f - mg$, which, under our hypotheses, have a positive right derivative except at the points of a countable subset of I.