266
III - Convergence: Continuous variables
Weierstrass in his turn tried his luck and produced the series L qn cos(anx),
normally convergent for 0 < q < 1, and proved that it has no derivative for
an odd integer 36 qa > 311"/2 + 1; just imagine the effect produced on mathematicians who, up until then, thought that a continuous function was always
differentiable except perhaps at a finite number of exceptional points, as was
"proved" by naIve diagrams that anyone can draw on a sheet of paper. But
no one has ever traced the graph of Weierstrass' function; you can of course
trace those of the partial sums of the series; computers having been invented
specifically to resolve this kind of practical problem, trajectories of a shell
(Eckert and Mauchly, 1942-1945) or missiles for example, you can probably
find the graphs on the Internet and download them onto your computer at
six o'clock in the morning when the majority of Americans are taking their
well earned rest after a hard day's work; but as to seeing what happens in
the limit ... This would anyway not help: Weierstrass proved his theorem
without waiting for the Web and, most likely, without wasting his time in
sketching graphs on graph paper.
What was not understood was that the situation is governed by the convergence of the derivatives and not by that of the given functions, as the
theory of integration will show yet more clearly:
Theorem 19. Let Un) be a sequence of functions defined and differentiable
on an interval I. Suppose that
(i) the sequence of derived functions U~) converges uniformly to a limit 9
on every compact K c I,
(ii) the sequence Un(x» converges at some point of I.
Then the functions In converge uniformly to a limit function I on every
compact K c I, and f' = g.
Choose acE I where limfn(c) exists. On replacing each function fn(x)
by fn(x)- fn(c), which does not change the derivatives and does not influence
the possible uniform convergence of In, one can assume that fn(c) = 0 for
all n. Put fpq(x) = fp(x) - Iq(x), whence fpq(c) = O. If K c I if; a compact
interval containing C and of length m(K) then Corollary 3 above shows that
(17.1 )
for all x E K. This can be written as
(17.2)
36 See Walter, Analysis 1, p. 359, for a full treatment of a similar example, due to
a Japanese at the beginning of the century (they "already" did mathematics,
and the principal collaborator of the physician and biologist Paul Ehrlich, who
discovered the first effective treatment for syphilis in about 1910, was also a
Japanese).
III - Convergence: Continuous variables
Weierstrass in his turn tried his luck and produced the series L qn cos(anx),
normally convergent for 0 < q < 1, and proved that it has no derivative for
an odd integer 36 qa > 311"/2 + 1; just imagine the effect produced on mathematicians who, up until then, thought that a continuous function was always
differentiable except perhaps at a finite number of exceptional points, as was
"proved" by naIve diagrams that anyone can draw on a sheet of paper. But
no one has ever traced the graph of Weierstrass' function; you can of course
trace those of the partial sums of the series; computers having been invented
specifically to resolve this kind of practical problem, trajectories of a shell
(Eckert and Mauchly, 1942-1945) or missiles for example, you can probably
find the graphs on the Internet and download them onto your computer at
six o'clock in the morning when the majority of Americans are taking their
well earned rest after a hard day's work; but as to seeing what happens in
the limit ... This would anyway not help: Weierstrass proved his theorem
without waiting for the Web and, most likely, without wasting his time in
sketching graphs on graph paper.
What was not understood was that the situation is governed by the convergence of the derivatives and not by that of the given functions, as the
theory of integration will show yet more clearly:
Theorem 19. Let Un) be a sequence of functions defined and differentiable
on an interval I. Suppose that
(i) the sequence of derived functions U~) converges uniformly to a limit 9
on every compact K c I,
(ii) the sequence Un(x» converges at some point of I.
Then the functions In converge uniformly to a limit function I on every
compact K c I, and f' = g.
Choose acE I where limfn(c) exists. On replacing each function fn(x)
by fn(x)- fn(c), which does not change the derivatives and does not influence
the possible uniform convergence of In, one can assume that fn(c) = 0 for
all n. Put fpq(x) = fp(x) - Iq(x), whence fpq(c) = O. If K c I if; a compact
interval containing C and of length m(K) then Corollary 3 above shows that
(17.1 )
for all x E K. This can be written as
(17.2)
36 See Walter, Analysis 1, p. 359, for a full treatment of a similar example, due to
a Japanese at the beginning of the century (they "already" did mathematics,
and the principal collaborator of the physician and biologist Paul Ehrlich, who
discovered the first effective treatment for syphilis in about 1910, was also a
Japanese).