Continuity
We have seen that any polynomial function
P
(
x
) satisfies:
for all real numbers
a
. This property is known as
continuity
.
Definition.
Let
f
(
x
) be a function defined on an interval around
a
. We say that
f
(
x
) is
continuous
at
a
iff
Otherwise, we say that
f
(
x
) is
discontinuous
at
a
.
Note that the continuity of
f
(
x
) at
a
means two things:
(i)
exists,
(ii)
and this limit is
f
(
a
).
So to be discontinuous at
a
, means
(i)
does not exist,
(ii)
or if
exists, then this limit is not equal to
f
(
a
).
Basic properties of limits imply the following:
Theorem.
If
f
(
x
) and
g
(
x
) are continuous at a. Then
(1)
f
(
x
) +
g
(
x
) is continuous at
a
;
(2)
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is continuous at
a
, where
is an arbitrary number;
(3)
is continuous at
a
;
(4)
is continuous at
a
, provided
;
(5)
If
f
(
x
) is positive, i.e.
, then
is continuous at
a
;
(6)
If
f
(
x
) is continuous at
a
and
g
(
x
) is continuous at
f
(
a
), then their composition
is continuous at
a
.
Remark.
Many functions are not defined on open intervals. In this case, we can talk about one
sided continuity. Indeed,
f
(
x
) is said to be
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 Spring '09
 Koong
 Calculus, Continuity, Continuous function

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