Given an open set $\Omega \subseteq \mathbb{R}^n$, a sequence of armonic functions $\{u_n\} \subseteq \Omega$ that are uniformly bounded on every compact set of $\Omega$ and that are supposed to reach almost everywhere a limit function $u$, what can we can say about the continuity of $u$?
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By Poisson the sequence is equicontinuous. By Ascoli ... – Daniel Fischer Dec 11 '20 at 20:45
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I can't get were you are taking me. I do not know what equicontinous funtion are and I do not know any Ascoli theorem. Can you please add more details or quote a reference? – Gabrielek Dec 11 '20 at 20:54
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This should get you started. The situation with harmonic functions is very similar to Montel's theorem. – Daniel Fischer Dec 11 '20 at 21:01