  Let $\Omega\subset\mathbb R^n$ be a $C^1$ domain, or more generally, a
Lipschitz domain with small local Lipschitz constant. In this paper it is shown
that if $u$ is a function harmonic in $\Omega$ and continuous in $\overline
\Omega$ which vanishes in a relatively open subset
$\Sigma\subset\partial\Omega$ and moreover the normal derivative $\partial_\nu
u$ vanishes in a subset of $\Sigma$ with positive surface measure, then $u$ is
identically $0$.
