Not necessarily- take any non-constant smooth function $w$ on $\mathbb{S}^2$ that vanishes on $\{x_3 = 0\}$, andlet let $\phi \in C^{\infty}_0(B_1)$ with$\Omega = B_1 \cap \{x_3 > 0\}.$ Then $\phi = 1$$u(x) := (1-|x|^2)\frac{x_3}{|x|}$ is in $B_{1/2}$. Then for $\Omega = B_1 \cap \{x_3 > 0\},$ we have
$$u(x) := \phi(x)w(x/|x|) \in H^1_0(\Omega) \cap C^{\infty}(\Omega)$$$H^1_0(\Omega) \cap C^{\infty}(\Omega),$
but $u$ is discontinuous at the origin.