Eigenfunction cannot have vanishing gradient along the boundary

Question

Let $\Sigma$ be a compact Riemannian surface with boundary and let $f \in C^{\infty}(\Sigma)$ be an eigenfunction of the Laplacian which vanishes on $\partial \Sigma$. How do I show that $\nabla f$ cannot vanish identically along $\partial \Sigma$? I think this is due to some sort of continuation principle, but I am no specialist.