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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a bounded [[open set|02BH3MCBQB]] with [[boundary|05D5EGMFYV]] $\\partial\\Omega$.\nFor a [[continuous function|0X3WGX95FE]] $f \\colon \\partial\\Omega \\to \\R$, let $H_f$ be its [[Perron–Wiener–Brelot solution|R12KQGVGN6]].\nA point $\\xi \\in \\partial\\Omega$ is a *regular boundary point* of $\\Omega$ if\n$$\\lim_{x \\to \\xi, \\, x \\in \\Omega} H_f(x) = f(\\xi)$$\nfor every continuous function $f$ on $\\partial\\Omega$.\nA boundary point that is not regular is *irregular*.\n\n## Remarks\n\nExample: if $\\Omega$ is the open unit ball, every boundary point is regular, because $H_f$ is the Poisson integral of $f$, which tends to $f(\\xi)$ at each $\\xi$.\nNon-example: for the punctured ball $\\Omega = \\set{0 \u003c \\abs{x} \u003c 1}$, the point $0$ is irregular: for $f = 0$ on the sphere and $f(0) = 1$, one has $H_f = 0$, which does not tend to $1$.\nEquivalently, $\\xi$ is regular if and only if there is a [[barrier|ED0SZS1VED]] at $\\xi$.\nThe Dirichlet problem on $\\Omega$ has a solution continuous on $\\overline{\\Omega}$ for every continuous $f$ if and only if every point of $\\partial\\Omega$ is regular; the solution is then $H_f$.\n",
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