260
III - Convergence: Continuous variables
All this may not be very clear at this level - it cannot be understood
outside the framework of finite dimensional vector spaces -, but one can
console oneself that the contemporaries and successors of Leibniz did not
understand it either. Whence the abundance of polemics and philosophical
dissertations on the subject until the appearance of "modern" mathematics
in the XX th century, of functions with vector values defined on a subset of
IR n , of linear and multilinear functions, etc. (Chap. IX, §1). This point of
view has been expounded with his habitual conciseness in Vol. I, Chap. VIII,
§ 12 of the Treatise on Analysis by Dieudonne, in Serge Lang, Analysis I,
Chap. XVI (same remark), and in many other works.
16 - The mean value theorem
Although the concept of a derivative was transformed into a formidable instrument by Newton, Leibniz and their successors, it had appeared earlier,
even though only implicitly, in Descartes, Fermat and Cavalieri. In Fermat
it is linked to seeking the maxima and minima of a function. This can be
'understood immediately, since, if f(x) ::::; f(c) for all x or even only on a
neighbourhood of c, then the ratio [f(c + h) - f(c)J/h which, in the limit,
defines the derivative at c, is ::::; 0 for h > 0 and 2: ° for h < 0 sufficiently
small: so it can tend only to 0, so long as f is defined on a neighbourhood of
c, and not only to the right or to the left of c.
As we saw above (nO 9, Corollary of Theorem 12), if we have a continuous function f with real values on a compact interval K = [a, b] then there
actually exists a point in K where the function attains a maximum (resp. a
minimum). This is not enough to prove that, if f is differentiable in K, its
derivative vanishes at these points: if f is monotone, for example linear, it
attains its minimum and its maximum at the end points of K and the preceding argument fails. To eliminate this objection it suffices to assume that
f(a) = feb); if one denotes by c' (resp. d') a point of K where f is a maximum (resp. a minimum), and if these two points are the end points of K, then
the function is everywhere between f(a) and feb) = f(a), so is constant, in
which case it is not difficult to prove that its derivative vanishes somewhere.
If, however, f is not constant, then at least one of the points d, c" is interior
to K, and the standard argument applies: f' vanishes somewhere between a
and b and even at a point c interior to K.
When f(a) f feb), one can reduce to the preceding case by replacing f(x)
by g(x) = f(x) + ux, where u is a constant chosen so that g(a) = g(b). This
requires f(a) + ua = feb) + ub, whence
g(x) = f(x) - [feb) - f(a)]x/(b - a).
Since g'(x) = f'(x) - [feb) - f(a)J1(b - a), we finally obtain the following
result:
III - Convergence: Continuous variables
All this may not be very clear at this level - it cannot be understood
outside the framework of finite dimensional vector spaces -, but one can
console oneself that the contemporaries and successors of Leibniz did not
understand it either. Whence the abundance of polemics and philosophical
dissertations on the subject until the appearance of "modern" mathematics
in the XX th century, of functions with vector values defined on a subset of
IR n , of linear and multilinear functions, etc. (Chap. IX, §1). This point of
view has been expounded with his habitual conciseness in Vol. I, Chap. VIII,
§ 12 of the Treatise on Analysis by Dieudonne, in Serge Lang, Analysis I,
Chap. XVI (same remark), and in many other works.
16 - The mean value theorem
Although the concept of a derivative was transformed into a formidable instrument by Newton, Leibniz and their successors, it had appeared earlier,
even though only implicitly, in Descartes, Fermat and Cavalieri. In Fermat
it is linked to seeking the maxima and minima of a function. This can be
'understood immediately, since, if f(x) ::::; f(c) for all x or even only on a
neighbourhood of c, then the ratio [f(c + h) - f(c)J/h which, in the limit,
defines the derivative at c, is ::::; 0 for h > 0 and 2: ° for h < 0 sufficiently
small: so it can tend only to 0, so long as f is defined on a neighbourhood of
c, and not only to the right or to the left of c.
As we saw above (nO 9, Corollary of Theorem 12), if we have a continuous function f with real values on a compact interval K = [a, b] then there
actually exists a point in K where the function attains a maximum (resp. a
minimum). This is not enough to prove that, if f is differentiable in K, its
derivative vanishes at these points: if f is monotone, for example linear, it
attains its minimum and its maximum at the end points of K and the preceding argument fails. To eliminate this objection it suffices to assume that
f(a) = feb); if one denotes by c' (resp. d') a point of K where f is a maximum (resp. a minimum), and if these two points are the end points of K, then
the function is everywhere between f(a) and feb) = f(a), so is constant, in
which case it is not difficult to prove that its derivative vanishes somewhere.
If, however, f is not constant, then at least one of the points d, c" is interior
to K, and the standard argument applies: f' vanishes somewhere between a
and b and even at a point c interior to K.
When f(a) f feb), one can reduce to the preceding case by replacing f(x)
by g(x) = f(x) + ux, where u is a constant chosen so that g(a) = g(b). This
requires f(a) + ua = feb) + ub, whence
g(x) = f(x) - [feb) - f(a)]x/(b - a).
Since g'(x) = f'(x) - [feb) - f(a)J1(b - a), we finally obtain the following
result: