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678 CHAPTER 19 Complex Numbers and Functions

If f is continuous at each point of a set S, we say that f is continuous on S. The function of
Example 19.7 is continuous on the set S consisting of the complex plane with i removed.

A function f is bounded on S if for some positive number M,
| f (z)|≤ M for all z in S.

This means that there must be a disk about the origin containing all the numbers f (z) for z in S.
A continuous function need not be bounded. For example, f (z) = 1/z is continuous on the
plane with the origin removed.
If, however, we place some conditions on the set S, then a function that is continuous on S
will be bounded on S.
One condition we will use is that the set is bounded (not to be confused with a function being
bounded). We say that a set S of complex numbers is bounded if, for some positive number K,
|z|≤ K for all z in S. This means that we can enclose all of S within a disk, if we choose the
radius large enough.

THEOREM 19.2

If f is continuous on a set S that is closed and bounded, then f is a bounded function on S.

A set that is closed and bounded is called compact. In this terminology, the theorem says
that a function that is continuous on a compact set must be a bounded function.

The complex derivative is modeled after the real derivative. We say that f is differentiable
at z 0 if for some number L
f (z) − f (z 0 )
lim = L.
z→z 0 z − z 0

This is equivalent to requiring that
f (z 0 + h) − f (z 0 )
lim = L
h→0 h
with the understanding that h is complex and must be allowed to approach 0 along an arbitrary
path.

In this case, we call L the derivative of f at z 0 and denote it f (z 0 ), or in the Leibniz notation,
d
f (z) .
dz
z=z 0
As with real functions, we rarely compute a complex derivative by applying the limit defini-
tion. The rules for computing derivatives of complex functions have the same form as those for
real functions whenever all of the derivatives are defined.

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October 15, 2010 18:5 THM/NEIL Page-678 27410_19_ch19_p667-694