506 17. REAL AND COMPLEX ANALYSISoperation still led to the invocation of some strong principles of inference in the
context of set theory. By mid-twentieth century mathematicians were accustomed
to proving concrete facts using abstract techniques. To take just one example, it
can be proved that some differential equations have a solution because a contraction
mapping of a complete metric space must have a fixed point. Classical mathemati-
cians would have found this proof difficult to accept, and many twentieth-century
mathematicians have preferred to write in "constructivist" ways that avoid invok-
ing the abstract "existence" of a mathematical object that cannot be displayed
explicitly. But most mathematicians are now comfortable with such reasoning.
2.6. Discontinuity as a positive property. The Weierstrass approximation
theorems imply that the property of being the limit of a sequence of continuous
functions is no more general than the property of being the limit of a sequence
of polynomials or the sum of a trigonometric series. That fact raises an obvious
question: What kind of function is the limit of a sequence of continuous functions?
As noted above, du Bois-Reymond had shown that it can be discontinuous on a set
that is, as we now say, dense. But can it, for example, be discontinuous at every
point? That was one of the questions that interested Rene-Louis Baire (1874-
1932). If one thinks of discontinuity as simply the absence of continuity, classifying
mathematical functions as continuous or discontinuous seems to make no more
sense than classifying mammals as cats or noncats. Baire, however, looked at the
matter differently. In his 1905 Lecons sur les functions discontinues (Lectures on
Discontinuous Functions) he wrote
Is it not the duty of the mathematician to begin by studying in
the abstract the relations between these two concepts of continuity
and discontinuity, which, while mutually opposite, are intimately
connected?
Strange as this view may seem at first, we may come to have some sympathy
for it if we think of the dichotomy between the continuous and the discrete, that is,
between geometry and arithmetic. At any rate, to a large number of mathemati-
cians at the turn of the twentieth century, it did not seem strange. The Moscow
mathematician Nikolai Vasilevich Bugaev (1837-1903, father of the writer Andrei
Belyi) was a philosophically inclined scholar who thought it possible to establish
two parallel theories, one for continuous functions, the other for discontinuous func-
tions. He called the latter theory arithmology to emphasize its arithmetic character.
There is at least enough of a superficial parallel between integrals and infinite se-
ries and between continuous and discrete probability distributions (another area
in which Russia has produced some of the world's leaders) to make such a pro-
gram plausible. It is partly Bugaev's influence that caused works on set theory
to be translated into Russian during the first decade of the twentieth century and
brought the Moscow mathematicians Luzin and Dmitrii Fyodorovich Egorov (1869-
1931) and their students to prominence in the area of measure theory, integration,
and real analysis.
Baire's monograph was single-mindedly dedicated to the pursuit of one goal:
to give a necessary and sufficient condition for a function to be the pointwise limit
of a sequence of continuous functions. He found the condition, building on earlier
ideas introduced by Hermann Hankel (1839-1873): The necessary and sufficient
condition is that the discontinuities of the function form a set of first category.