§l. The intermediate value theorem
197
We observe in passing an apparently strange consequence of the preceding
Fourier series: a series whose terms are everywhere continuous functions of
a real variable x and which converges everywhere may have a discontinuous
function for its sum. We shall return to this very important problem in nO 5
and 6. For the moment we only remark that Fourier's formula assumes that
his series converges; the proof, unfortunately, is not entirely obvious. We shall
give it at the end of nO 11 since it provoked one of the first efforts to ground
analysis on a solid base, and applies to many similar series.
Most of the engineers and technicians who use these series find no problem: the result is part of the folklore of the profession to the same extent
as the dangers of alternating current. The "high fidelity" journals sometimes
speak without explanation of the unpropitious influence of the "odd harmonics" on the fidelity of the hi-fi: this impresses the punters to the same extent
as the mysterious "RMS power" (root mean square) defined by an integral
which we shall meet again Ii propos Fourier series.
3 - Right and left limits of a monotone function
Let f be a real function defined on a subset X of R We say that f is increasing
(in the wide sense) if, for x,y E X, the relation x ~ y implies f(x) ~ f(y);
on substituting strict for wide inequalities we obtain the strictly increasing
functions; this is the case of log x as we saw in Chap. II, nO 10, and also
of the function exp since exp(x + h) = exp(x) exp(h) > expx if h > 0 (use
the power series in h). We define decreasing and strictly decreasing functions
similarly. When a function is increasing or decreasing one may say that it
is monotone, not specifying more precisely which. When X is an interval, it
may happen that one can decompose X into subintervals so that the function
f, while not monotone on all of X, is monotone on each ofthese subintervals;
one says then that f is piecewise monotone in X. This is the case for all
the elementary functions whose graph one requires the future graduates to
trace on millimeter graph paper; but contemplating these drawings, all the
less realistic as they are more artistic, would never give an idea of the level
of complexity a monotone function can attain; you will find an example in
Chap. V, nO 32 of an increasing function which is discontinuous at all the
points of Ql and continuous elsewhere.
Now consider a real function, defined and increasing on a subset X of IR,
and let c be an adherent point of X; let us write E for the set of x E X such
that x < c (strict inequality) and confine ourselves to considering f on E,
supposing that E is nonempty and that c is still adherent to E. When x E E
tends to c while remaining < c, the values taken by f(x) increase (in the
wide sense); the arguments of Chap. II, nO 9 now suggest that f(x) tends
to a limit value, maybe +00. Arguing as in Chap. II, we are led to consider
all the numbers M which majorise f(x) for all x < c and to show that f(x)
converges to the least of these numbers (or to +00 if no such exists), namely
u = sup(f(E)) ~ +00, the least upper bound of the set f(E) of values taken
197
We observe in passing an apparently strange consequence of the preceding
Fourier series: a series whose terms are everywhere continuous functions of
a real variable x and which converges everywhere may have a discontinuous
function for its sum. We shall return to this very important problem in nO 5
and 6. For the moment we only remark that Fourier's formula assumes that
his series converges; the proof, unfortunately, is not entirely obvious. We shall
give it at the end of nO 11 since it provoked one of the first efforts to ground
analysis on a solid base, and applies to many similar series.
Most of the engineers and technicians who use these series find no problem: the result is part of the folklore of the profession to the same extent
as the dangers of alternating current. The "high fidelity" journals sometimes
speak without explanation of the unpropitious influence of the "odd harmonics" on the fidelity of the hi-fi: this impresses the punters to the same extent
as the mysterious "RMS power" (root mean square) defined by an integral
which we shall meet again Ii propos Fourier series.
3 - Right and left limits of a monotone function
Let f be a real function defined on a subset X of R We say that f is increasing
(in the wide sense) if, for x,y E X, the relation x ~ y implies f(x) ~ f(y);
on substituting strict for wide inequalities we obtain the strictly increasing
functions; this is the case of log x as we saw in Chap. II, nO 10, and also
of the function exp since exp(x + h) = exp(x) exp(h) > expx if h > 0 (use
the power series in h). We define decreasing and strictly decreasing functions
similarly. When a function is increasing or decreasing one may say that it
is monotone, not specifying more precisely which. When X is an interval, it
may happen that one can decompose X into subintervals so that the function
f, while not monotone on all of X, is monotone on each ofthese subintervals;
one says then that f is piecewise monotone in X. This is the case for all
the elementary functions whose graph one requires the future graduates to
trace on millimeter graph paper; but contemplating these drawings, all the
less realistic as they are more artistic, would never give an idea of the level
of complexity a monotone function can attain; you will find an example in
Chap. V, nO 32 of an increasing function which is discontinuous at all the
points of Ql and continuous elsewhere.
Now consider a real function, defined and increasing on a subset X of IR,
and let c be an adherent point of X; let us write E for the set of x E X such
that x < c (strict inequality) and confine ourselves to considering f on E,
supposing that E is nonempty and that c is still adherent to E. When x E E
tends to c while remaining < c, the values taken by f(x) increase (in the
wide sense); the arguments of Chap. II, nO 9 now suggest that f(x) tends
to a limit value, maybe +00. Arguing as in Chap. II, we are led to consider
all the numbers M which majorise f(x) for all x < c and to show that f(x)
converges to the least of these numbers (or to +00 if no such exists), namely
u = sup(f(E)) ~ +00, the least upper bound of the set f(E) of values taken