The name is absent

THE RISE OF FUNCTIONS                  11

whether an analytic function of t has to be indeed analytic throughout, or
may be only piecewise analytic in finitely many adjoining intervals
[4, pp. 287-288].

At first glance, the formation of a functional object by putting together
pieces of analytic or other “well-defined” functions may appear to be,
mathematically, a makeshift operation, an ingenuous one. But the 19th
century learned to respect, explore, and exploit such formations; and in the
20th century there would hardly be any topology if it were not for Simplicial
and related decompositions and approximations.

Trigonometric Series. Amidst all its uncertainties about the nature of a
function, the 18th century somehow managed to make the capital discovery
— which, in a way, has been unmatched since — that functions of a very
“general” class can be represented in the form

OO

(5)            /(-x)=⅛Λq+ ∑ (u,,cosmx + b_πsinnx).

n= 1

In the early 19th century, Fourier greatly emphasized what had been
known before, that for a given/(x) the corresponding “Fourier coefficients”
in the expansion (5) usually have the values

If»                   1 f“ .

(6)      a,, = — /(x)cosnxrfx, b„ = — I /(x)sinnxrfx,

π J-_π                    π

and he also greatly emphasized that “any” function/(x) has a representation
(5), even if the function is absolument arbitraire.

As it turned out, this absolument arbitraire was a great “challenge”
(à la Toynbee), and, in a sense, the creation of the theory of functions of a
real variable was a “response” to this challenge.

Firstly, Dirichlet made the following contributions (1829-1837):

(i) He gave his famed “definition” of a truly “arbitrary” numerical
function y = /(x), as a “general” correspondence from x to y.

(ii) He introduced — perhaps for the first time — a specific class of
functions of a real variable to a specific purpose. It was the class of piecewise
monotone functions; and Dirichlet established the fact that for such a
function the Fourier series converges at all points.

(After the rise of set theory, towards the end of the 19th century, these
functions of Dirichlet “engendered” functions of bounded variation and
also rectifiable curves.)

Secondly, Riemann made the following contributions (1854, published
1867):

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