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a
in thedomain of thefunction
f
,
(ii) If for every sequence (
a
)
with
a
a
,(
f
(
a
))
l
, provethat
lim
f
(
x
)
l
.
89 If, given
0, there exists a
such that when
x
belongs to the
domain of
f
and 0
x
a
f
(
x
)
f
(
a
)
, provethat
when
x
belongs to the domain of
f
and
x
a
, then
f
(
x
)
, then
f
(
a
)
.
Deduce that if lim
f
(
x
)
f
(
a
) then
f
is continuous at
a
.
Now we have established that a function
f
is continuous at a point
a
of its domain if and only if
f
(
a
).
lim
f
(
x
)
90 Find a valueof
k
such that thefunction
f
: R
R defined by
2
x
k
when
x
1,
f
(
x
)
when
x
1,
is continuous at
x
1.
R
91 If thefunction
f
:[
a
,
b
]
is continuous at each point of its
domain givethevalueof
f
(
x
) for
c
lim
f
(
x
), lim
f
(
x
) and lim
(
a
,
b
).
92 If
f
(
a
), lim
f
(
b
) and
lim
f
(
x
)
f
(
x
)
lim
f
(
x
)
f
(
c
), for every
c
(
a
,
b
),
provethat
f
is continuous at every point of [
a
,
b
]. Useqn 89, and
consider sequences for the one-sided limits.
Theorems on limits
93
The algebra of limits
Use theorems about sequences to prove that if
l
and lim
m
,
lim
f
(
x
)
g
(
x
)
