Let $U$ be an open connected subset of $\mathbb{C}$ and let $u:U\rightarrow \mathbb{R}$ be harmonic and bounded on $U$.

Let $f:\partial_\infty U \rightarrow \mathbb{R}$ be a continuous function, and suppose $u$ extends continuously to $f$ on a dense subset of $\partial_\infty U$.

By $\partial_\infty U$ I mean the boundary of $U$ in $\mathbb{C}\cup\{\infty\}$. By "$u$ extends continuously to $f$ on $D$" I mean that for each $a \in D$, $\lim_{z\rightarrow a} u(z)$ exists and is equal to $f(a)$.

Does it follow that $u$ extends continuously to $f$ on all of $\partial_\infty U$?