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.

strictly increasing function on an interval of positive length, such that for all . Let be the image of and let

**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

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

(14.45) |

(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.