1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 567

Proof L'et M be the set of such zeros. We have f(M) = {O} and M =

f-^1 {O} n D. Since {O} is closed and f is continuous, it follows that the

set M is closed in D (Theorem 3.7).

Theorem 8.19 Let f: D -t C ,he continuous with D open. If f(z)

vanishes in a set everywhere dense in D, then f(z) = 0 for all z ED.

Proof The continuity of f implies that A = D is the set of zeros of f

whenever f(A) = {O} and A is everywhere dense in D.
Corollary 8.9 Let f: D -t C and g: D -t C be continuous with D open,
and let A be a set everywhere dense in D. If f(z) = g(z) for all z E A,
then f(z) = g(z) for all z ED; i.e., a continuous function is determined by
its values on a set everywhere dense in its domain of definition.

Proof The function F(z) = f(z)-g(z) is continuous on D and vanishes in

a set everywhere dense in D. Hence F(z) = 0 or f(z) = g(z) for all z ED.

Remarks I. We have shown in Theorem 8.18 that the set M of zeros
of finite order of a continuous function in an open set D is closed in D.

Conversely,· if M C D is closed, there is always a continuous function f

such that f(z) = 0 if z E M and f(z) =f. 0 if z E D - M.

Example Let

f(z) = d(z, M) = glb(EM lz - Cl

Then f(z) = 0 iff z EM. Clearly, f is continuous on M. To show that f is

also continuous at any point zo ED -M, first we note that glbCEM lzo -Cl
is attained at some point C' E M. From

z - C' = ( z - zo) + ( zo -(')

we obtain

Jz -('J::; Jz - zol + lzo - ('J

= lz -zoJ + glb(EM lzo - Cl

and for some other z E D - M,

glhceM lz -Cl::; Jz - C'I::; Jz -zol + glhceM lzo -CJ

so that

f(z) - f(zo)::; Jz - zoJ

Similarly,

f(zo) - f(z)::; Jz - zol