Graduate Is There a Boundary Point Where the Holomorphic Function Equals Zero?

[JYM]

I want to solve the following problem:
Suppose B=B(0,R) be a ball in C^n, n>1. Let f be holomorphic in B and continuous on B closure. If f(a)=0 for some a in B, show that there is p in boundary of B such that f(p)=0.
I assumed f(p) is non zero for every point p in boundary B and create contradiction but I can't. Please give me some hints.

 

[mathwonk]

a function that is holomorphic in an open neighborhood of a sphere (boundary of a ball) in C^n, n > 1, extends to be holomorphic in the interior of that ball. in your case, if the function is never zero on the boundary, then its reciprocal is holomorphic in an open neighborhood of the boundary of a slightly smaller ball, hence also in the interior of that smaller ball. now try to get a contradiction. this uses a theorem in chapter 1 of gunning and rossi, analytic functions of several complex variables, p.20, the section I.C on removable singularities. of course you also need the identity (uniqueness) principle for holomorphic functions.

 

[JYM]

Ok, Thanks.