- REAL ANALYSIS 503
2.3. Uniform convergence and continuity. Cauchy was not aware at first of
any need to make the distinction between pointwise and uniform convergence, and
he even claimed that the sum of a series of continuous functions would be contin-
uous, a claim contradicted by Abel, as we have seen. The distinction is a subtle
one. It is all too easy not to notice whether choosing ç large enough to get a good
approximation when fn(x) converges to f(x) requires one to take account of which
÷ is under consideration. That point was rather difficult to state precisely. The first
clear statement of it is due to Philipp Ludwig von Seidel (1821-1896), a professor
at Munich, who in 1847 studied the examples of Dirichlet and Abel, coming to the
following conclusion:
When one begins from the certainty thus obtained that the propo-
sition cannot be generally valid, then its proof must basically lie
in some still hidden supposition. When this is subject to a precise
analysis, then it is not difficult to discover the hidden hypothesis.
One can then reason backwards that this [hypothesis] cannot oc-
cur [be fulfilled] with series that represent discontinuous functions.
[Quoted in Bottazzini, 1986, p. 202]In order to reason confidently about continuity, derivatives, and integrals, math-
ematicians began restricting themselves to cases where the series converged uni-
formly. Weierstrass, in particular, provided a famous theorem known as the M-test
for uniform convergence of a series. But, although the M-test is certainly valuable
in dealing with power series, uniform convergence in general is too severe a restric-
tion. The important trigonometric series studied by Abel, for example, represented
a discontinuous function as the sum of a series of continuous functions and there-
fore did not converge uniformly. Yet it could be integrated term by term. One
could provide many examples of series of continuous functions that converged to a
continuous function but not uniformly. Weaker conditions were needed that would
justify the operations rigorously without restricting their applicability too strongly.
2.4. General integrals and discontinuous functions. The search for less re-
strictive hypotheses and the consideration of more general figures on a line than
just points and intervals led to more general notions of length, area, and integral,
allowing more general functions to be integrated. Analysts began generalizing the
integral beyond the refinements introduced by Riemann. Foundational problems
also added urgency to this search. For example, in 1881, Vito Volterra (1860-1940)
gave an example of a continuous function having a derivative at every point, but
whose derivative was not Riemann integrable. What could the fundamental the-
orem of calculus mean for such a function, which had an antiderivative but no
integral, as integrals were then understood?
New integrals were created by the Latvian mathematician Axel Harnack (1851-
1888), by the French mathematicians Emile Borel (1871-1956), Henri Lebesgue
(1875-1941), and Arnaud Denjoy (1884-1974), and by the German mathematician
Oskar Perron (1880-1975). By far the most influential of these was the Lebesgue
integral, which was developed between 1899 and 1902. This integral was to have
profound influence in the area of probability, due to its use by Borel, and in trigono-
metric series representations, an application that Lebesgue developed, perhaps as
proof of the usefulness of his highly abstruse integral, which, as a former colleague
of the author was fond of saying, "did not change any tables of integrals." Lebesgue