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10.2 Differentiable Functions on

10.21   Warning. By the definition of differentiablity given in Math 111, the domain of a function was required to contain some interval in order for the function be differentiable at . In definition 10.1 this condition has been replaced by requiring to be a limit point of the domain of the function. Now a function whose domain is a closed interval may be differentiable at and/or .

10.22   Definition (Critical point.) Let be a complex function, and let . If is differentiable at and , we call a critical point for .

10.23   Theorem (Critical Point Theorem.) Let be a function. Suppose has a maximum at some point , and that contains an interval where . If is differentiable at , then . The theorem also holds if we replace  maximum" by  minimum."

Proof: Suppose has a maximum at ,

Define two sequences , in by

Clearly and , and and for all . We have

By the inequality theorem,

Also,

so

Since , we conclude that . The proof for minimum points is left to you.

10.24   Theorem (Rolle's Theorem.) Let with and let
be a function that is continuous on and differentiable on . Suppose that . Then there is a number such that .

Proof: We know from the extreme value theorem that has a maximum at some point . If , then the critical point theorem says , and we are finished. Suppose . We know there is a point such that has a minimum at . If we get by the critical point theorem, so suppose . Then since and , we have , and it follows that is a constant function on , and in this case for all .

10.25   Theorem (Mean Value Theorem.) Let with , and let be a function from to such that is continuous on and differentiable on . Then there is a point such that

This theorem says that the tangent to the graph of at some point is parallel to the chord joining to .

Proof: Let

so the equation of the line joining to is , and

Let

Then

and is continuous on and differentiable on . By Rolle's theorem, there is some such that ; i.e., ; i.e.,

10.26   Remark. The mean value theorem does not hold for complex valued functions. Let

Then

so

But

so , and there is no point in with

10.27   Definition (Interior point.) Let be an interval in . A number is an interior point of if and only if is not an end point of . The set of all interior points of is called the interior of and is denoted by .

10.28   Examples. If , then

If is an interval, and are points in with , then every point in is in the interior of .

10.29   Theorem. Let be an interval in , and let be a continuous function on . Then:
a)
If for all , then is increasing on .
b)
If for all , then is strictly increasing on .
c)
If for all , then is decreasing on .
d)
If for all , then is strictly decreasing on .
e)
If for all , then is constant on .

Proof: All five statements have similar proofs. I'll prove only part a).

Suppose for all . Then for all with we have is continuous on and differentiable on , so by the mean value theorem

Hence, is increasing on .

10.30   Exercise. Prove part e) of the previous theorem; i.e., show that if is an interval in and is continuous and satisfies for all , then is constant on . [It is sufficient to show that for all .]

10.31   Exercise. A For each assertion below, either prove that the assertion is true for all functions , or give a function for which the assertion is false. (A proof may consist of quoting a theorem.)

a) If is differentiable on and is strictly increasing on , then for all .

b) If is differentiable on , and has a maximum at , then .

c) If is continuous on and is differentiable on , and for all , then is strictly increasing on .

10.32   Theorem (Restriction theorem)

Let be a subset of , let , and let be a point such that is differentiable at . Let be a subset of containing , and let be the restriction of to , i.e.

If is a limit point of , then is differentiable at , and

Proof: Let be any sequence in such that . Then is a sequence in , and hence

It follows that

I've shown that

10.33   Definition (Path, line segment.) If , then the path joining to is the function

and the set

is called the line segment joining to .

10.34   Example. We showed in example 10.9 that the function is a nowhere differentiable function on . I will show that for all , in with , the restriction of conj to the line segment is differentiable, and

Note that all points of are limit points of . If , then for some real number
 (10.35)

and
 (10.36)

If we solve equation (10.35) for we get

By using this value for in equation (10.36) we get

Let be defined by

Then is differentiable, and for all . We have

so by the restriction theorem

10.37   Exercise. A Let denote the unit circle in . Show that is differentiable, and that

In general, the real and imaginary parts of a differentiable function are not differentiable.

10.38   Example. If for all , then is differentiable and . However, is nowhere differentiable. In fact, if , has no limit at . To see this, let for all . Then , , and

and

Hence, the sequences and have different limits, so does not exist.

However, we do have the following theorem.

10.39   Theorem. Let be an interval in and let be a function differentiable at a point . Write where are real valued. Then and are differentiable at , and .

Proof: Since is differentiable at there is a function on such that is continuous at and

If and , then and , so
 (10.40)

and
 (10.41)

Since is continuous at , and are continuous at , so equations (10.40) and (10.41) show that and are differentiable and

10.42   Example. Let , and let for all . Then is differentiable and (by the chain rule),

We have by direct calculation,

so

(This example just illustrates that the theorem is true in a special case.)

10.43   Theorem. Let be a complex function and let , and suppose contains the line segment , and that for all . Then is constant on ; i.e., for all .

Proof: Define a function by

By the chain rule, is differentiable on and . Since for all , we have for all . Hence

and hence

If and , then for all .

10.44   Exercise. Let be a disc in .
a)
Show that if then the segment is a subset of .
b)
Let be a function such that for all . Show that is constant on .

Next: 10.3 Trigonometric Functions Up: 10. The Derivative Previous: 10.1 Derivatives of Complex   Index