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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 $\partial_\infty U$?

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Take a dense open set of small measure on the circle, take $u$ to be the Poisson extension of its characteristic function, and take $f=1$. –  fedja May 24 '11 at 23:09
    
Thank you. I only knew of the Poisson extension for continuous functions, but it seems that it works for $L^p$ more generally. Is there a standard reference for the Poisson extension for $L^p$ functions? –  Linda Brown Westrick May 27 '11 at 2:07
    
@Linda A standard reference is Gilbarg's and Trudinger's "Elliptic Partial Differential Equations of Second Order." You want to look at the strong solution chapter. –  Yakov Shlapentokh-Rothman May 27 '11 at 16:55

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