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worthy of consideration. Following his study of series of trigonometric
functions, Fourier insisted (1822) that the values of a function need not
be smoothly connected. In 1829 Dirichlet proposed a function like that
of qn 20 in his discussion of Fourier series and integration and it was in
1837 that Dirichlet gave what amounts to the modern definition of
function in which each value of
x
gives a unique value of
f
(
x
), which
need not depend in any way on the value given for a different
x
, and
the function is defined by the pairing (
x
,
f
(
x
)).
Thenotation
f
(
x
)or
(
x
) dates from Euler (1734) though separating
the variable in brackets only became standard practice after Cauchy
(1821).
Until the early years of the twentieth century a real function might
takethevalue
, so that, for example, if
f
(
x
)
1/
x
,
f
(0)
. In 1820,
Herschel used the symbol
f
for theinvrsefunction, but this did not
become standard practice until the twentieth century. Describing
functions in terms of their domains and co-domains followed the
set-theoretic formulation of mathematics in the early years of the
twentieth century. The terms injection, surjection and bijection were
devised by the twentieth century corporate French mathematician
Bourbaki.
It has always been accepted that an equation like
x
y
1
determines a functional relationship between
y
and
x
. But a given value
of
x
between
1 gives two possible values of
y
, since
y
x
(1
).
Such
many
-
valued
functions appear in the literature right through the
nineteenth century, and functions were taken to be
single
-
valued
only
when they were said to be so. However for the purposes of defining
continuity, limits, and inverse functions, single-valued functions are
essential and so during the twentieth century
functions
havecometo
mean
single
-
valued
functions, though this convention is still not
universal.
In order to formulate the Intermediate Value Theorem both
Bolzano (1817) and Cauchy (1821) needed a definition of continuity,
and said that
f
was continuous when
f
(
x
h
)
f
(
x
) became small with
h
. Their definitions are recognisable as our neighbourhood definition of
continuity though they only considered functions which were
continuous at least on an interval, not just at an isolated point. For his
proof of the Intermediate Value Theorem, Bolzano needed the result
that a function which is continuous and takes a non-zero value at a
point must be non-zero in a neighbourhood of that point. The first
description of a limit appears in Cauchy's description of
derivative (1823) without symbolic use of absolute values or inequalities.
The
definition of limit was formulated by Weierstrass using the
modern notation for absolute value, which he introduced, in his first
