- REAL ANALYSIS 505
geometric figures, it was necessary to investigate differentiation in more detail as
well.
The secret of that quest turned out to be not continuity, but monotonicity.
A continuous function may fail to have a derivative, but in order to fail, it must
oscillate very wildly, as the examples of Bolzano and Weierstrass did. A function
that did not oscillate or oscillated only a finite total amount, necessarily had a
derivative except on a set of measure zero. The ultimate result in this direction
was achieved by Lebesgue, who showed that a monotonic function has a derivative
on a set whose complement has measure zero. Such a function might or might not
be the integral of its derivative, as the fundamental theorem of calculus states. In
1902 Lebesgue gave necessary and sufficient conditions for the fundamental theorem
of calculus to hold; a function that satisfies these conditions, and is consequently
the integral of its derivative, is called absolutely continuous.
To return to the problem of abstractness, we note that it had been known
at least since the time of Lagrange that any finite set of ç data points (xk,yk),
k = É,.,.,ç, with Xk all different, could be fitted perfectly with a polynomial
of degree at most ç — 1. Such a polynomial might—indeed, probably would—
oscillate wildly in the intervals between the data points. Weierstrass showed in
1884 that any continuoiis function, no matter how abstract, could be uniformly
approximated by a polynomial over any bounded interval [a, b\. Since there is always
some observational error in any set of data, this result meant that polynomials could
be used in both practical and theoretical ways, to fit data, and to establish general
theorems about continuous functions. Weierstrass also proved a second version of
the theorem, for periodic functions, in which he showed that for these functions the
polynomial could be replaced by a finite sum of sines and cosines. This connection
to the classical functions freed mathematicians to use the new abstract functions,
confident that in applications they could be replaced by computable functions.
Weierstrass lived before the invention of the new abstract integrals mentioned
above arose, although he did encourage the development of the abstract set theory
of Georg Cantor on which these integrals were based. With the development of
the Lebesgue integral a new category of functions arose, the measurable functions.
These are functions f(x) such that the set of ÷ for which f(x) > c always has a
meaningful measure, although it need not be a geometrically simple set, as it is in
the case of continuous functions. It appeared that Weierstrass' work needed to be
repeated, since his approximation theorem did not apply to measurable functions.
In his 1915 dissertation Luzin produced two beautiful theorems in this direction.
The first was what is commonly called by his name nowadays, the theorem that for
every measurable function f(x) and every å > 0 there is a continuous function g(x)
such that g(x) ö /(÷) only on a set of measure less than å. As a consequence of this
result and Weierstrass' approximation theorem, it followed that every measurable
function is the limit of a sequence of polynomials on a set whose complement has
measure zero. Luzin's second theorem was that every finite-valued measurable
function is the derivative of a continuous function at the points of a set whose
complement has measure zero. He was able to use this result to show that any
prescribed set of measurable boundary values on the disk could be the boundary
values of a harmonic function.
With the Weierstrass approximation theorem and theorems like those of Luzin,
modern analysis found some anchor in the concrete analysis of the "classical" pe-
riod that ran from 1700 to 1900. But that striving for generality and freedom of