- Suppose that \(\Omega\) is a simply connected domain that is bounded by a piecewisesmooth closed curve \(\gamma\) (in the terminology of Chapter 1). Then any conformal map \(F\) of D to Ω extends to a continuous bijection of \(\overline{\mathbb D}\) to \(\overline{\Omega}\). The proof is simply a generalization of the argument used in Theorem 4.2.
Solution
The proof of Theorem 4.2 for polygonal regions uses only two geometric facts about the target boundary: the region is bounded, and for every boundary point \(w_0\) there is \(r_0>0\) such that for \(0<r<r_0\) the intersection \[ \Omega\cap \{w:|w-w_0|=r\} \] is a single arc. We verify that a piecewise-smooth Jordan boundary has the same local property, and then the proof of Theorem 4.2 applies verbatim.
Let \(w_0=\gamma(t_0)\). At a smooth point, \(\gamma'(t_0)\ne0\) and \[ \gamma(t)=w_0+\gamma'(t_0)(t-t_0)+o(|t-t_0|). \] After a rotation and translation, the curve is therefore locally the graph of a continuous function over its tangent line, with the two sides of the Jordan curve lying on opposite sides. Hence every sufficiently small circle centered at \(w_0\) meets \(\gamma\) in exactly two nearby points, and its intersection with \(\Omega\) is the single open arc between them. At a corner the same conclusion follows from the two one-sided nonzero tangent vectors. Since the boundary is piecewise smooth and simple, this gives the required local-arc property at every boundary point.
Now repeat the proof of Theorem 4.2. For \(\zeta_0\in\partial\mathbb D\), let \(C_r\) be the circle centered at \(\zeta_0\) of radius \(r\). If \(z_r,z_r'\in C_r\cap\mathbb D\) are joined by the arc of \(C_r\cap\mathbb D\), then Cauchy–Schwarz gives \[ |F(z_r)-F(z_r')|^2 \le 2\pi r\int_{C_r\cap\mathbb D}|F'(z)|^2\,|dz|. \] If the left side were bounded below by a positive constant for every sufficiently small \(r\), integration in \(dr/r\) would give an infinite lower bound, while \[ \iint_{\mathbb D}|F'(z)|^2\,dx\,dy =\operatorname{Area}(\Omega)<\infty, \] a contradiction. Thus there is a sequence \(r_n\downarrow0\) along which the oscillation of \(F\) on the relevant circular arc tends to zero.
Suppose now that \(F(z)\) had two different cluster values \(w,w'\in\partial\Omega\) as \(z\to\zeta_0\). Choose disjoint small discs around \(w,w'\). By the local-arc property of \(\Omega\), each disc meets \(\Omega\) in a connected arc-like neighborhood, so one can join the corresponding subsequences of image points to \(w\) and \(w'\) by curves lying in those disjoint neighborhoods. Pulling these curves back by \(F^{-1}\) produces two curves in \(\mathbb D\) approaching \(\zeta_0\). Every sufficiently small \(C_r\) meets both curves, at points \(z_r,z_r'\), but their images stay a fixed positive distance apart. This contradicts the preceding oscillation estimate along the sequence \(r_n\). Therefore \[ \lim_{z\to\zeta_0,\,z\in\mathbb D}F(z) \] exists for every \(\zeta_0\in\partial\mathbb D\).
Define the boundary values by these limits. The same sequential argument used in Theorem 4.2 shows that the extension \[ \overline F:\overline{\mathbb D}\to\overline\Omega \] is continuous.
Apply the identical argument to the conformal inverse \(G=F^{-1}:\Omega\to\mathbb D\). The local-arc property just verified for \(\Omega\) replaces the polygonal boundary property used in Theorem 4.2, while the unit circle has the corresponding property automatically. Hence \(G\) extends continuously to \[ \overline G:\overline\Omega\to\overline{\mathbb D}. \] For \(z\in\partial\mathbb D\), choose \(z_n\in\mathbb D\) with \(z_n\to z\). Since \(G(F(z_n))=z_n\), continuity gives \[ \overline G(\overline F(z))=z. \] Similarly \(\overline F(\overline G(w))=w\) for \(w\in\partial\Omega\). Thus the two extensions are inverse homeomorphisms. In particular, \(F\) extends to a continuous bijection \[ \boxed{\overline{\mathbb D}\xrightarrow{\sim}\overline\Omega}. \]