§4. Differentiable functions
261
Theorem 18 (mean value theorem). Let f be a real junction defined
and differentiable on an interval I. For any a, bEl there exists a number
c E la, b[ such that
(16.1)
f(b) - f(a) = (b - a)f'(c).
The geometric interpretation is obvious: there exists a point with abscissa c where the tangent to the graph of f is parallel to the "chord", of
slope [f(b) - f(a)]j(b - a), joining the points of the graph with abscissae a
and b. If one puts b = a + h in (1) one finds
(16.1 ')
f(a + h) = f(a) + f'(a + th)h
where t (traditionally denoted 0) is a number lying between 0 and 1 and of
course depending on a and hj compare with the relation f(a + h) = f(a) +
f'(a)h + o(h). A curious aspect of (1') is that, when h tends to 0, the term
f'(a + th) tends to f'(a)j since 0 < t < 1, this would be obvious if f' were
continuous at the point a, but we have not made this hypothesisj this shows
that the points a + th which appear in (1 '), for example for h = lin, are not
distributed at random ...
On this theme we remark that if f' is not continuous, it is still not a very
savage function - just enough to be well beyond the imagination of people who
are not experts in set theory Ii la Cantor (Georg) and Ii la Baire: assuming
for simplicity that f is defined and differentiable on all JR, the function f' is
a simple limit of continuous junctions, namely of
fn(x) = n(f(x + lin) - f(x)].
Reread the end of nO 6.
We can generalise formula (1) by modifying the argument a little.
Instead of looking for a function of the form f(x) + ux which takes the
same values at a and b, let us choose any differentiable function g( x) and
consider the function f(x) + ug(x) = h(x). We will have h(a) = h(b) if
U = -[f(b) - f(a)]j[g(b) - g(a)]. Since h'(x) = f'(x) + ug'(x) and since h'
vanishes at some point c E la, b[, we have f'(c) + ug'(c) = 0, and hence the
formula
(16.1")
f(b) - f(a)
g(b) - g(a)
f'(c)
g'(c)
which, for g(x) = x, reduces to Theorem 18.
There is no result analogous to (1) for a complex-valued function: there is
of course a c' for its real part and a C" for its imaginary part, but why should
they be equal? In this case one replaces the formula (1) by an inequality - see
below - at least as useful in practice. First, let us note the more important
consequences of Theorem 18.
261
Theorem 18 (mean value theorem). Let f be a real junction defined
and differentiable on an interval I. For any a, bEl there exists a number
c E la, b[ such that
(16.1)
f(b) - f(a) = (b - a)f'(c).
The geometric interpretation is obvious: there exists a point with abscissa c where the tangent to the graph of f is parallel to the "chord", of
slope [f(b) - f(a)]j(b - a), joining the points of the graph with abscissae a
and b. If one puts b = a + h in (1) one finds
(16.1 ')
f(a + h) = f(a) + f'(a + th)h
where t (traditionally denoted 0) is a number lying between 0 and 1 and of
course depending on a and hj compare with the relation f(a + h) = f(a) +
f'(a)h + o(h). A curious aspect of (1') is that, when h tends to 0, the term
f'(a + th) tends to f'(a)j since 0 < t < 1, this would be obvious if f' were
continuous at the point a, but we have not made this hypothesisj this shows
that the points a + th which appear in (1 '), for example for h = lin, are not
distributed at random ...
On this theme we remark that if f' is not continuous, it is still not a very
savage function - just enough to be well beyond the imagination of people who
are not experts in set theory Ii la Cantor (Georg) and Ii la Baire: assuming
for simplicity that f is defined and differentiable on all JR, the function f' is
a simple limit of continuous junctions, namely of
fn(x) = n(f(x + lin) - f(x)].
Reread the end of nO 6.
We can generalise formula (1) by modifying the argument a little.
Instead of looking for a function of the form f(x) + ux which takes the
same values at a and b, let us choose any differentiable function g( x) and
consider the function f(x) + ug(x) = h(x). We will have h(a) = h(b) if
U = -[f(b) - f(a)]j[g(b) - g(a)]. Since h'(x) = f'(x) + ug'(x) and since h'
vanishes at some point c E la, b[, we have f'(c) + ug'(c) = 0, and hence the
formula
(16.1")
f(b) - f(a)
g(b) - g(a)
f'(c)
g'(c)
which, for g(x) = x, reduces to Theorem 18.
There is no result analogous to (1) for a complex-valued function: there is
of course a c' for its real part and a C" for its imaginary part, but why should
they be equal? In this case one replaces the formula (1) by an inequality - see
below - at least as useful in practice. First, let us note the more important
consequences of Theorem 18.