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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be an open set, let $u$ be a [[superharmonic function|X4V2S56YQQ]] on $\\Omega$, and let $x \\in \\Omega$.\nThen there is a set $E \\subseteq \\Omega \\setminus \\set{x}$ that is [[thin|7DGF6MMEGX]] at $x$ such that\n$$\\lim_{y \\to x,\\ y \\in \\Omega \\setminus E} u(y) = u(x).$$\n\n## Remarks\n\nEquivalently, $u$ is continuous at $x$ for the [[fine topology|W2ZD4XP74R]], once the finely open sets are described as the sets whose complement is thin at each of their points.\nIdea of the proof: if $u(x) = +\\infty$, lower semicontinuity already gives $u(y) \\to +\\infty$ as $y \\to x$, and one takes $E = \\emptyset$.\nIf $u(x) \u003c \\infty$, then for each $\\varepsilon \u003e 0$ the set $\\set{y : u(y) \u003e u(x) + \\varepsilon}$ is thin at $x$, with $u$ itself as the witness.\nThese sets, cut down to suitable shrinking balls about $x$, combine into a single set $E$ that is still thin at $x$.\nThis step uses the description of thinness by capacity.\nExample: in $\\R^3$, the potential $u(y) = \\sum_{k \\ge 1} 4^{-k} \\abs{y - 2^{-k} e_1}^{-1}$ is superharmonic with $u(0) = 1$ and $u = +\\infty$ at the points $2^{-k} e_1$.\nSo $u$ is not continuous at $0$.\nIt tends to $1$ as $y \\to 0$ outside the union $E$ of the balls of radius $k^2 4^{-k}$ about the points $2^{-k} e_1$, by dominated convergence.\nThis $E$ is thin at $0$ by Wiener's criterion, since $\\sum_k 2^k k^2 4^{-k} \u003c \\infty$.\nThe exceptional set cannot be dropped in general, as this example shows.\n\n## Explanations\n\n### L1 human\n\nA superharmonic function can jump up at points arbitrarily close to $x$.\nThe theorem says that these jumps happen only on a set that is negligible near $x$ in the sense of potential theory.\nOutside that set, the function behaves continuously at $x$.\n\n### L2 human\n\nThis is the result that makes the fine topology useful.\nIt shows that the topology defined abstractly by superharmonic functions has a concrete description by thin sets, so fine limits can be checked with capacities through Wiener's criterion.\nFine limits then appear in the boundary theory of harmonic functions, for instance in the fine form of Fatou-type theorems on the Martin boundary.\n",
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