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  "body": "\n## Statement\n\nLet $n \\ge 3$ and let $\\mu$ be a positive [[Radon measure|HZNEQQBCW4]] on $\\R^n$ with compact [[support|FQTXV6BQZG]] $S$.\nLet $U^\\mu(x) = \\int \\abs{x - y}^{2-n} \\, d\\mu(y)$ be its [[Newtonian potential|DCMFQ01QGS]].\nLet $x_0 \\in S$, and assume that the restriction of $U^\\mu$ to $S$ is continuous at $x_0$, with $U^\\mu(x_0) \u003c \\infty$.\nThen $U^\\mu$, as a function on all of $\\R^n$, is continuous at $x_0$.\n\n## Remarks\n\nThis result is also called the Evans–Vasilesco theorem.\nProof idea: split $\\mu$ into the part $\\mu_1$ on a small ball $B(x_0, \\delta)$ and the rest $\\mu_2$.\nThe potential of $\\mu_2$ is continuous near $x_0$, since its kernel is continuous there.\nFor $x$ near $x_0$, let $y$ be a point of $S$ nearest to $x$; every $z \\in S$ satisfies $\\abs{y - z} \\le 2 \\abs{x - z}$, so $U^{\\mu_1}(x) \\le 2^{n-2} U^{\\mu_1}(y)$.\nThe continuity on $S$ and $U^\\mu(x_0) \u003c \\infty$ make $U^{\\mu_1}$ small on $S$ near $x_0$ when $\\delta$ is small, hence also off $S$, and lower semicontinuity of $U^\\mu$ gives the other inequality.\n\nExample: for the uniform measure $\\sigma$ of mass $1$ on the unit sphere in $\\R^3$, the potential equals $1$ on the sphere, so its restriction to the support is continuous.\nThe theorem then says that $U^\\sigma$ is continuous on $\\R^3$, and indeed $U^\\sigma(x) = \\min(1, 1/\\abs{x})$.\n\nWithout the hypothesis on the restriction, the potential can be discontinuous at a point of finite potential.\nIn $\\R^3$, fix a ray from $x_0$ and let $c_k$ be its point at distance $2^{-k}$ from $x_0$.\nLet $\\sigma_k$ be the uniform probability measure on the sphere of radius $4^{-k}$ centred at $c_k$, and let $\\mu = \\sum_{k \\ge 2} k \\, 4^{-k} \\sigma_k$.\nThen $U^\\mu(x_0) \\le \\sum_k k \\, 4^{-k} / (2^{-k} - 4^{-k}) \u003c \\infty$.\nBut $U^\\mu(c_k) \\ge k$, and $c_k \\to x_0$.\nThe same statement holds for logarithmic potentials in the plane.\n\n## Explanations\n\n### L1 human\n\nA potential is always lower semicontinuous, but it may jump up near points where the charge is concentrated.\nThe continuity principle says that such jumps can only be seen from the charge itself.\nIf the potential behaves continuously when one stays on the charge, it behaves continuously when one comes from any direction.\n\n### L2 human\n\nThe principle reduces questions of continuity of potentials to the support of the measure.\nIt is used in the proof of the maximum principle for potentials, to pass from inequalities on the support to inequalities everywhere.\n",
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