  Let $u$ be a harmonic function in a $C^1$ domain $D\subset \mathbb{R}^d$,
which vanishes on an open subset of the boundary. In this note we study its
critical set $\{x \in \overline{D}: \nabla u(x) = 0 \}$. When $D$ is a
$C^{1,\alpha}$ domain for some $\alpha \in (0,1]$, we give an upper bound on
the $(d-2)$-dimensional Hausdorff measure of the critical set by the frequency
function. We also discuss possible ways to extend such estimate to all
$C^1$-Dini domains, the optimal class of domains for which analogous estimates
have been shown to hold for the singular set $\{x \in \overline{D}: u(x) = 0 =
|\nabla u(x)| \}$ (see [KZ1, KZ2]).
