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14.38
Lemma.
Let be a strictly increasing continuous function whose domain is
an interval . Then the image of is the interval
, and the function
has an inverse.
Proof: It is clear that and are in image.
Since is continuous we can apply the intermediate value property
to conclude that for every number between and
there is a number such that ,
i.e.
image. Since is increasing
on we have
whenever
,
and thus image
. It follows that
is surjective, and since strictly increasing
functions are injective, is bijective. By remark (14.22)
has an inverse.
14.39
Exercise.
State and prove the analogue of lemma
14.38
for strictly decreasing functions.
14.40
Exercise.
Let
be a function whose domain
is an interval
, and whose image is an interval. Does it
follow that
is continuous?
14.41
Exercise.
A
Let
be a continuous function on a
closed bounded interval
. Show that the image of
is a closed bounded interval
.
14.42
Exercise.
Let
and
be non-empty intervals and let
be a
continuous function such that
image(
).
a) Show that if is strictly increasing, then the inverse function
for is also strictly increasing.
b) Show that if is strictly decreasing, then the inverse function for
is also strictly decreasing.
Remark:
If is a nonvertical line joining two points and
then the slope of is
The reflection of about the line whose equation is passes
through the points and , so the slope of the reflected
line is
Thus theorem 14.43 says that the tangent to graph() at the
point is obtained by reflecting the tangent to graph()
at about the line whose equation is . This is what you should
expect from
the geometry of the situation.
Proof of theorem 14.43:
The first thing that should be done, is to prove that is continuous.
I am going to omit that proof and just assume the continuity of ,
and then show that is differentiable,
and that is given by formula (14.44).
Let be a point
in the interior of
. then
(Observe that we have not divided by zero). Let
be a sequence in
such that .
Then
(since is assumed to be
continuous), and
for all
(since is injective). Since is differentiable
at , it follows that
Since
it follows that
It follows that
and the theorem is proved.
Remark: The inverse function theorem also applies to continuous
functions on such that for all in interior .
Formula (14.44) is valid in this case also.
Remark: Although we have stated the inverse function
theorem for functions on intervals of the form , it holds for
functions defined on any interval. Let be an interval, and let
be a continuous strictly increasing function from to R such that
for all in the interior of . Let be a point
in the interior of image. Then we can find points and
in image such that . Now maps the interval
bijectively onto , and since
we can apply the inverse function theorem on the interval
to conclude that
. It is not necessary
to remember the formula for . Once we know that is
differentiable, we can calculate by using the chain rule, as illustrated
by the examples in the next section.
Next: 14.6 Some Derivative Calculations
Up: 14. The Inverse Function
Previous: 14.4 The Exponential Function
  Index
Ray Mayer
2007-09-07