§2. Uniform convergence
215
ideas and his formulations are even so close to those of Bolzano that in 1975
I. Grattann-Guinness tried, comparing the texts, to prove that Cauchy had
plagiarised Bolzano, a theory refuted by Hans Freudenthal (especially in his
superb article on Cauchy in the DSB) and hardly plausible in view of the
obscurity in which Bolzano's activities were then surrounded: he was not, by
far, a full time professional integrated into the "international mathematical
community". In particular he seems to have been the first to reflect on the
concept of the least upper bound of a set; he showed that one can approximate it indefinitely closely by rational numbers, but, alas, was unable to
prove its existence, which continued to be impossible so long as one lacked
an exact definition of the real numbers. These ideas, that one also meets at
the same period with Gauss, were manifestly in the air at the time; but the
problem, solved by Dedekind in the years from 1858, held up the attempts
to arithmetise analysis for as long as one spoke of numbers whose nature one
had not understood.
All this poses the very difficult problem of characterising by internal properties - nature of the discontinuities - the functions which are simple limits
of continuous functions. It was solved by Rene Baire in his doctoral thesis
(1899) by using extraordinarily subtle techniques of set theory in R., which -
including the plain statement of the result - go much beyond the level of the
present exposition; Baire's methods paved the way for many French works
(Borel, Lebesgue, Denjoy), then gave rise, between the two wars, to Russian
and Polish works, more general, or analogous, and often more useful. The
reader may console himself for not knowing this famous theorem of Baire's
on the limits of continuous functions by reading what Dieudonne justifiably
wrote to Dugac in 1976:
with the benefit of hindsight it is quite clear today that the greater part
of these works [of Baire, Borel, Lebesgue and Denjoy] came to a dead end;
essentially there remain "Baire's theorem" 12 and the Lebesgue integral,
two fundamental tools in all analysis; but all the rest, for the moment
at Least, are museum pieces; I have never seen a problem (not concocted
ad hoc) where the famous theorem on "pointwise discontinuous" functions
[Le. the characterisation by Baire of simple limits of continuous functions]
arises anywhere.
12 Let us say that a set X C JR is everywhere dense in a set Y C JR if every point
of Y is adherent to X (so a limit of points of X, example Q in JR). Then the
intersection of a countable family of open everywhere dense sets in JR is again
everywhere dense in JR. On taking complements one obtains the following statement: if Fn is a sequence of closed sets not containing any interior point, neither
does their union. There are other formulations, some applicable to spaces much
more general than lR. See Dieudonne, Treatise on Analysis, Vol. 2, Chap. XII,
nO 16.
215
ideas and his formulations are even so close to those of Bolzano that in 1975
I. Grattann-Guinness tried, comparing the texts, to prove that Cauchy had
plagiarised Bolzano, a theory refuted by Hans Freudenthal (especially in his
superb article on Cauchy in the DSB) and hardly plausible in view of the
obscurity in which Bolzano's activities were then surrounded: he was not, by
far, a full time professional integrated into the "international mathematical
community". In particular he seems to have been the first to reflect on the
concept of the least upper bound of a set; he showed that one can approximate it indefinitely closely by rational numbers, but, alas, was unable to
prove its existence, which continued to be impossible so long as one lacked
an exact definition of the real numbers. These ideas, that one also meets at
the same period with Gauss, were manifestly in the air at the time; but the
problem, solved by Dedekind in the years from 1858, held up the attempts
to arithmetise analysis for as long as one spoke of numbers whose nature one
had not understood.
All this poses the very difficult problem of characterising by internal properties - nature of the discontinuities - the functions which are simple limits
of continuous functions. It was solved by Rene Baire in his doctoral thesis
(1899) by using extraordinarily subtle techniques of set theory in R., which -
including the plain statement of the result - go much beyond the level of the
present exposition; Baire's methods paved the way for many French works
(Borel, Lebesgue, Denjoy), then gave rise, between the two wars, to Russian
and Polish works, more general, or analogous, and often more useful. The
reader may console himself for not knowing this famous theorem of Baire's
on the limits of continuous functions by reading what Dieudonne justifiably
wrote to Dugac in 1976:
with the benefit of hindsight it is quite clear today that the greater part
of these works [of Baire, Borel, Lebesgue and Denjoy] came to a dead end;
essentially there remain "Baire's theorem" 12 and the Lebesgue integral,
two fundamental tools in all analysis; but all the rest, for the moment
at Least, are museum pieces; I have never seen a problem (not concocted
ad hoc) where the famous theorem on "pointwise discontinuous" functions
[Le. the characterisation by Baire of simple limits of continuous functions]
arises anywhere.
12 Let us say that a set X C JR is everywhere dense in a set Y C JR if every point
of Y is adherent to X (so a limit of points of X, example Q in JR). Then the
intersection of a countable family of open everywhere dense sets in JR is again
everywhere dense in JR. On taking complements one obtains the following statement: if Fn is a sequence of closed sets not containing any interior point, neither
does their union. There are other formulations, some applicable to spaces much
more general than lR. See Dieudonne, Treatise on Analysis, Vol. 2, Chap. XII,
nO 16.