Does the Maximum Principle Apply to Harmonic Functions in Bounded Regions?

[haya]

Homework Statement

Let u(x; y) be real, nonconstant, and continuous in a closed
bounded region R. Let u(x; y) be harmonic in the interior of R. Prove that
the maximum and minimum value of u(x; y) in this region occurs on the boundary.

Homework Equations

the theorem said that( a function analtic in bounded domain and continuous up to and including its boundary attains its maximum modlus on the boundary

The Attempt at a Solution

can i suppose that u(x;y) is nonzero ?

 

[ninty]

Is R connected?
If so, this is just the maximum principle