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Let $\Omega\subset\mathbb R^n$ be a $C^1$ domain, or more generally, aLipschitz domain with small local Lipschitz constant. In this paper it is shownthat 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 $\partial_\nuu$ vanishes in a subset of $\Sigma$ with positive surface measure, then $u$ isidentically $0$.

Unique continuation at the boundary for harmonic functions in $C^1$ domains and Lipschitz domains with small constant