Nonconstant harmonic function that is constant on open subset The following question arose while I was doing some homework - it's not a homework problem, just a question that might allow me to generalize the homework problem:

Let $h$ be a (real-valued) nonconstant harmonic function on an open connected set $U$. Is it possible that for some nonempty open subset of $U$ that $h$ is constant?
Reference: Theorems 1.27 and 1.28 (page 21) of Harmonic Function Theory by Sheldon Axler, Paul Bourdon, and Wade Ramsay.
No, by the isolated zeroes theorem.