Graphics Reference
In-Depth Information
Summary
-
continuity by sequences
The sequence definition of continuity
qn 18
A function
f
:
A
R is said to be
continuous
at
a
A
when, for
every
sequence (
a
)
a
with
))
f
(
a
).
Theorem
If thefunctions
f
:
A
R and
g
:
A
R areboth
continuous at
a
A
, then
terms in
A
,(
f
(
a
qn 41
(i)
f
is continuous at
a
;
qn 52
(ii) 1/
f
is continuous at
a
, provided
f
(
a
)
0;
qn 23
(iii)
f
g
is continuous at
a
;
qn 26
(iv)
f
ยท
g
is continuous at
a
;
qn 54
(v)
f
/
g
is continuous at
a
, provided
g
(
a
)
0.
Definition
qn 38
If thefunction
f
is defined on the range of the
function
g
, then the function of
f
g
is defined by
f
g
(
x
)
f
(
g
(
x
))
If thefunction
g
is continuous at thepoint
a
,
and thefunction
f
is continuous at thepoint
g
(
a
), then the composite function
f
g
is
continuous at thepoint
a
.
A squeeze rule
If for three functions
f
,
g
and
h
qn 36
Theorem
qn 40
(i)
f
(
x
)
g
(
x
)
h
(
x
) in a neighbourhood of the
point
a
;
(ii)
f
(
a
)
h
(
a
);
(iii)
f
and
h
arecontinuous at thepoint
a
;
then the function
g
is continuous at
a
.
Theorem
qn 21
A constant function is continuous at each point
of its domain.
Theorem
qn 22
The identity function is continuous at each
point of its domain.
Theorem
qn 29
A polynomial function is continuous at each
point of its domain.
Neighbourhoods
Wesay that a function
f
is continuous at a point
x
a
when '
f
(
x
)
tends to
f
(
a
), as
x
tends to
a
', which wemay think of as meaning 'as
x
gets near to
a
,
f
(
x
) gets near to
f
(
a
)'. So far, we have always described
nearness
using sequences and convergence. Another way to describe
nearness is to consider
neighbourhoods
of a point
a
.
56 Illustratethests
x
:
1
x
3
,
x
:
2
x
1
2
,
x
:
x
1
2