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  "body": "\n## Statement\n\nLet $\\mathbb{D}$ be the open [[unit disc|8G5ABXRK85]].\nLet $J \\subset \\C$ be a Jordan curve, that is, a subset homeomorphic to a circle, and let $\\Omega$ be the bounded component of $\\C \\setminus J$.\nLet $f \\colon \\mathbb{D} \\to \\Omega$ be a [[conformal map|0VHNKP1N3K]] of $\\mathbb{D}$ onto $\\Omega$.\nThen $f$ extends to a [[homeomorphism|05GZ11PWRY]] of the [[closure|0DRKT1QJR7]] $\\overline{\\mathbb{D}}$ onto $\\overline{\\Omega} = \\Omega \\cup J$.\n\n## Remarks\n\nSuch an $f$ exists by the [[Riemann mapping theorem|02WHMEG5ZR]], since $\\Omega$ is simply connected by the [[Jordan curve theorem|687RGCHBTP]].\nThe proof rests on a length–area estimate.\nLet $\\zeta$ be a point of the unit circle and let $r$ be small.\nBy the Cauchy–Schwarz inequality and the finite area of $\\Omega$, some arc $\\set{\\abs{z - \\zeta} = \\rho} \\cap \\mathbb{D}$ with $r^2 \u003c \\rho \u003c r$ has an image of length at most $C/\\sqrt{\\log(1/r)}$.\nBecause $J$ is a Jordan curve, a short crosscut of $\\Omega$ cuts off a piece of small diameter, and this gives the continuity of $f$ at $\\zeta$.\nThe Jordan property also gives injectivity on the circle.\nThe converse holds: if $f$ extends to such a homeomorphism, the boundary of $\\Omega$ is the image of the circle, hence a Jordan curve.\nA more general form says that $f$ extends continuously to $\\overline{\\mathbb{D}}$ if and only if the boundary of $\\Omega$ is locally connected.\nExample: the boundary of the slit disc $\\mathbb{D} \\setminus [0, 1)$ is not a Jordan curve.\nA conformal map from $\\mathbb{D}$ onto it extends continuously, but two points of the circle go to each inner point of the slit, so the extension is not injective.\n\n## Explanations\n\n### L1 human\n\nThe Riemann mapping theorem gives a conformal map from the disc onto any proper simply connected region, but it says nothing about the edge.\nCarathéodory's theorem says that if the edge of the region is a simple closed curve, the map behaves well up to the edge.\nThe circle is then mapped continuously and one-to-one onto the boundary curve.\n\n### L2 human\n\nThe theorem lets one transfer boundary value problems, such as the [[Dirichlet problem|8D56GQ4AG8]], from the disc to any Jordan domain.\nIt also shows that a conformal map between two Jordan domains extends to a homeomorphism of their closures, by composing two Riemann maps.\nFor domains with worse boundaries, the right replacement for boundary points is the notion of a [[prime end|GNWE3XR0ZG]].\nCarathéodory's prime end theorem gives a bijection between the circle and the prime ends.\n",
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