The Development of Continuity
Dr. Paul Ehrlich
February, 2002
Abstract -- Why do we have the precise, but hard to learn at first
glance, delta-epsilon definition of continuity and differentiability in
our calculus texts ? During much of the 1800's, the most popular texts
were free of this bothersome unpleasantness. The answer is partly
that work by Joseph Fourier in his 1807 manuscript on the heat equation
(using what are now called Fourier series to solve that equation)
contradicted widely accepted (but erroneous) results for infinite series
of functions
which had been obtained without the full precision of the delta-epsilon
approach and related concepts such as uniform convergence.
1. Fourier against the establishment
Here is how the concept of continuity was defined in one of the
standard calculus texts of the 1800's, S. F. Lacroix's Traite
elementaire de calcul differentiel et de calcul integral --
as translated in [B, p. 91] --
By the law of continuity is meant that which is observed in the description
of lines by motion, and according to which the consecutive points of the
same line succeed each other without any interval.
This definition was first given in the first edition of this text, published
in 1802, and remained unchanged even up through the sixth edition, which was
published in 1858.
In this same general time frame, the eminent French mathematician
Augustin-Louis Cauchy was active as the Professor of Mathematics at the
eminent French engineering academy in Paris, the Ecole Royale Polytechnique.
As such, Cauchy had the duty of giving a lecture course in calculus to the
student body. As is done even today in France, it was customary for lecture
notes of such courses to be produced (nowdays by graduate students) and so
it is possible to see what Cauchy himself taught by looking in his
Oeuvres completes contained in any University library.
With definitions such as the above imprecise notion to work with, it is
entirely forgivable if errors were made enroute to the development of the
rigorous calculus.
Here is one of the results Cauchy gave in his lecture course, hence it must
have been widely believed. Let
f1(x), f2(x), ..., fn(x), ...
be a countable sequence of continuous functions. Then the sum
S(x) = f1(x) + f2(x) + ... +
fn(x) + ...
is continuous.
We teach nowdays in calculus that the sum of two (and hence any finite number)
of continuous functions is always continuous, but here Cauchy was asserting
that this finite result could be extended to any COUNTABLY infinite number
of functions !
The notational machinery with which Cauchy formulated his proof would be
familiar to any reader of a current calculus text. Let Sn(x)
denote the nth partial sum
Sn(x) = f1(x) + f2(x) + ... +
fn(x)
and let Rn(x) denote the remainder
Rn(x) = S(x) - Sn(x) = fn + 1(x)
+ fn + 2(x) + ....
What is 100 % correct is that Sn(x) is a continuous function for
any n as a finite sum of continuous functions.
As transcribed by Bressoud [B, p. 187], Cauchy gave the following proof of
the result
extended to infinite sums of continuous functions :
Let us consider the changes in these three functions when we increase x by
an infinitely small value alpha. For all possible values of n, the change
in Sn(x) will be infinitely small; the change in Rn(x)
will be as insignificant (or insensible) as the size of Rn(x)
when n is made very large. It follows that the change in the function
S(x) can only be an infinitely small quantity. From this remark we
immediately deduce the following proposition:
THEREOM I -- When the terms of a series are functions of a single variable
x and are continuous with respect to this variable in the neighborhood of a
particular value where the series converges, the sum S(x) of the series is
also, in the neighborhood of this particular value, a continuous function of
x.
According to Bressoud, Cauchy's claimed proof of this result can be understood
in the following way. Since S(y) = Sn(y) + Rn(y) for
any y in |R, we may derive the following inequality:
|S(x) - S(a)| = |(Sn(x) + Rn(x)) - (Sn(a) +
Rn(a))|
= |(Sn(x) - Sn(a)) + Rn(x)
- Rn(a)|
which is by the triangle inequality, less than or equal to
(*) |Sn(x) - Sn(a)| + |Rn(x)|
+ |Rn(a)|.
Now make the first term abitrarily small by the continuity of Sn
(taking x close to a) and make the second two terms arbitrarily small by
taking n as large as needed.
In December, 1807, Joseph Fourier submitted a manuscript for publication to
the Institut de France in Paris, which in translation, had the title
Theory of Propagation of Heat in Solid Bodies. Applying the separation
of variables method to the heat equation (e.g., assuming that there is a
solution of the form z = f(x) g(t) ), Fourier obtained solutions to the
heat equation with initial condition 1 for -1 < x < 1 which he wrote in terms
of what we now call Fourier series . A particular example is given
by (in the choice in the above notation) of
fn(x) = (-1)n + 1(4/pi) cos((2n - 1)pi x / 2)
/(2n - 1)
and S(x) the associated infinite sum. Clearly if any functions fn
(x) are continuous for all real numbers, those formed out of cosines certainly
are. Thus, according to the result of Cauchy's calculus lectures, the
associated function S(x) should be continuous for all real numbers. However,
Fourier's analysis of this function (cf. [Bressoud, p. 7] for a graph)
revealed that it was a function of period 2 with jump discontinuities at the
odd integers. Thus, for example, S(x) = 1 for -1 < x < 1, but then S(x) =
-1 for 1 < x < 3, so S jumps from the value +1 to the value -1 at x = 1.
Such behavior, of course, precludes continuity. On these (and other) grounds,
scientists were reluctant to accept Fourier series for several decades.
In Bressoud, [B, pp. 188 - 190], a nice telescoping series is used to
illustrate what goes wrong with Cauchy's analysis of (*) above. We choose
fk(x) = x2/[(1 + kx2)
(1 + (k - 1)x2)]
which by partial fraction techniques may also be written as
fk(x) = [1/(1 + (k -1)x2] - [
1/(1 + kx2)].
Accordingly, the partial sums Sn(x) display the telescoping behavior
beloved of calculus textbook writers, and we have
(**) Sn(x) = 1 - [1/(1 + nx2)
hence, also,
(***) Sn(x) = [nx2]/[1 + nx2].
From (**), we have for any nonzero x that Sn(x) approaches 1 as
n tends to + infinity, thus S(x) = 1 for any nonzero x. But from (***),
we have Sn(0) = 0 for any n, so that S(0) = 0. Hence the
power series S(x) with kth term defined as above converges for all x, but
is discontinuous at x = 0, since S(x) = 1 for x nonzero, but S(0) = 0.
We are now in a position to investigate the terms of decomposition (*)
in this particular example, with a = 0. Recall that Sn
(0) = S(0) = 0, so that Rn(0) = 0 also. We obtain the expression
|Sn(x) - Sn(0)| + |Rn(x)| +
|Rn(0)| = |Sn(x) - 0| + |Rn(x)|
+ |0|
= |Sn(x)| + |Rn(x)|
(****) = [nx2/(1 + nx2)] +[1/(1 +nx2)]
Intuitively, Cauchy's idea in this example is to make the first term in (****)
arbitrarily small by taking x close to zero. But then notice that the
second term in the expression (****) becomes arbitrarily close to 1,
unless n can then be taken larger and larger. But doing that undoes the work
accomplished in making the first term small by making x small, given n.
From a slightly different viewpoint, we have bounded |S(x) - S(0)| by
expression (****) in carrying out the Cauchy argument, but in this particular
case, one can even observe by doing the algebra that (****) = 1, hence no
juggling of x and n whatsoever will make (****) arbitrarily small and the
proof fails.
What we see emerging here is the need for the concept of uniform
convergence (which was not widely appreciated until the 1860's),
and indeed, nowdays we teach in Advanced Calculus the result that if the
infinite series S(x) converges uniformly in an interval (a,b) and
each of
the summands fn(x) is continous at every point of the interval
(a,b), then the series S is itself continuous at every point of (a,b).
But what became of the Fourier manuscript, submitted for publication in
December, 1807 ? Bressoud in his book reveals what happened at the beginning
of Chapter 6 ([B, pp. 219]):
"In the spring of 1808, Simeon Dennis Poisson wrote up the committee's report
on Fourier's Theory of Propagation of Heat in Solid Bodies. The
conclusion was that it contained nothing that was new or interesting. Behind
this opinion lay Lagrange's opposition to the admission of Fourier's
trigonometric series and his conviction that they must not converge. In the
following years, Fourier attempted to meet Lagrange's objections and to
convince him that his series did in fact converge. In the meantime, he
conducted experiments, comparing the predictions of his mathematical models
with observed phenomena.
The problem of modeling the flow of heat was of concern to many scientists
of the time. In 1811, the Institut de France announced a competition for the
best explanation of heat diffusion. Fourier reworked his earlier manuscript
and submitted it. Despite continuing objections from Lagrange, he was
awarded the prize. Lagrange could not deny him the award, but he could
postpone publication. Even after Lagrange died in 1813, Fourier's manuscript
continued to languish at the Institut. Fourier began to prepare a book to
disseminate his ideas.
After the second fall of Napoleon, Fourier came back to Paris as the Director
of the Bureau of Statistics for the department of the Seine. He was back at
the center of intellectual life. His book, Theorie analytique de la
chaleur (Analytic theory of heat), appeared in 1822. That same year
he was elected perpetual secretary of the Academie des Sciences, the highest
of scientific honors. He used that position in succeeding years to encourage
and promote the careers of emerging mathematicians. Gustav Dirichlet, Sophie
Germain, Joseph Liouville, Claude Navier, and Charles Sturm were among those
who received his assistance and would remember him fondly."
2. The Special Case of Power Series
In the case of a power series, in the notation of the previous section, we
have
fn(x) = cn (x - a) n.
Also, in our calculus texts, we encounter the following result on
differentiation and integration of power series, term by term (cf. [S,
pp. 746 - 747]) --
THEOREM 2 -- If the power series Summation { cn
(x - a)
n} has radius of convergence R > 0, then the function
f(x) = co + c1(x - a) + c2 (x - a)
2+ ... + cn (x - a)n +
...
is differentiable (and therefore continuous) on the interval (a - R,a + R)
and
(i) f '(x) = c1 + 2c2(x - a) + 3c3
(x - a)2 + ...
(ii) Integral{ f(x) dx } = C + co(x - a) + c1
(x - a)2/2 + ....
The radii of convergence of the power series in Equations (i) and (ii) are
both R.
Unfortunately, for series of more general functions fn(x) than
cn (x - a)n, the question of when term-by-term
differentiation is permissable is much more complicated, and indeed, such
questions were at the heart of some of the questioning of the legitimacy
of Fourier's methods when his
paper was being reviewed in 1807 - 1808 before this theory had been well
understood or even developed.
Here is a typical result from the general theory (cf. [B, pp. 199]):
Theorem -- Let F(x) = f 1(x) + f 2(x) +
... + fn(x) + ... be an infinite series convergent at
x = a and such that each fk(x) is differentiable at every
point in an open interval I containing x = a. If
Summation { fk '(x) }
converges uniformly over the interval I, then F(x) is differentiable
at x = a and
F '(a) = f1'(a) + f2'(a) + ... +
fn'(a) + ...
For power series, with f k(x) = ck (x - a) k,
and the wonderful elementary limit
lim (k) 1/k = 1 as k tends to + infinity
one has
lim sup (|ck|1/k) = lim sup ((k |c
k|)1/k).
Hence, the radius of convergence of any power series is the same as the radius
of convergence of the series of derivatives. At each point inside the interval
of absolute convergence, we are inside an interval at which the series and
series of derivatives both converge uniformly. Hence, the conditions required
for "good" behavior of a series of functions, as given in the above sample
Theorem, are always automatically satisfied in the case of power series, and
the more complicated behavior of Fourier series, which was so troubling to the
19th century mathematicians, was not hinted at from their earlier
understanding of power series.
References
1). Bressoud, David, A Radical Approach to Real Analysis, Mathematical
Association of America, Classroom Resource Materials Series, Vol. 2,
1994.
2). Stewart, James, Calculus, Early Transcendentals , Fourth
Edition, Brooks/Cole Publishing Company, New York, 1999.