= Caratheodory boundary extension theorem
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For a bounded <simply connected domain> $D$ and a <conformal bijection> $f:\mathbb D\to D$, $f$ extends continuously to the closed disc if and only if $\partial D$ is a <locally connected space>. A Jordan <boundary> gives a homeomorphism of closed discs, but <local connectedness> alone need not give injectivity on the <boundary>: a slit has two <boundary> approaches. The same statement applies on the sphere to unbounded <domains> after a suitable change of coordinates. This <boundary> theorem is different from the <Caratheodory extension theorem> for measures.
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