Graphics Reference
In-Depth Information
l
if and only if, given
Theorem
qn 88
lim
0, there
exists a
such that when
x
is in thedomain of
f
and 0
f
(
x
)
x
a
, then
f
(
x
)
l
.
Theorem
qns 75, 78,
89
The definition of continuity by limit
Thefunction
f
is continuous at
a
if and only if
lim
f
(
x
)
f
(
a
).
Theorem
qn 93
If lim
f
(
x
)
l
and lim
g
(
x
)
m
, then
(i) lim
f
(
x
)
l
;
g
(
x
)
l
m
,
(ii) lim
f
(
x
)
(iii) lim
f
(
x
) ·
g
(
x
)
l
·
m
,
(iv) lim
1/
f
(
x
)
1/
l
, provided
l
0.
T
heorem
qn 98
If for three functions
f
,
g
and
h
(i)
f
(
x
)
g
(
x
)
h
(
x
) except possibly when
x
a
,
(ii) lim
f
(
x
)
l
and lim
h
(
x
)
l
.
then lim
g
(
x
)
l
.
The completeness principle has not been used in this chapter, so all the
definitions and theorems here are valid just with Archimedean order.
Historical note
Throughout the seventeenth and eighteenth centuries, and for many
mathematicians well on into the nineteenth century, the only functions
which were under consideration were polynomials, rational functions,
trigonometric and exponential functions, and combinations of these.
Although the verbal definitions of a function used during the eighteenth
century sound like modern definitions, only functions defined by a
formula were under consideration. Such functions are continuous and
infinitely differentiable except possibly at isolated points (e.g. at
x
0
for thefunction
x
1/
x
). Even continuous functions given by different
formulae on different segments of their domain, such as the equation of
a plucked string, were only admitted with discomfort.
It was the consideration of limiting processes by Fourier and
Cauchy, among others, which began to widen the field of functions
