Generalization of the argument used in theorem

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The results of given Exercise extend to the case when γ is assumed merely to be closed, simple, and continuous. The proof, however, requires further ideas.

Exercise

Suppose that ? is a simply connected domain that is bounded by a piecewise smooth closed curve γ (in the terminology of Chapter 1). Then any conformal map F of D‾ to ?‾ extends to a continuous bijection of D to ?. The proof is simply a generalization of the argument used in Theorem.

Theorem
If F: D → P is a conformal map, then F extends to a continuous bijection from the closure D‾ of the disc to the closure P‾ of the polygonal region. In particular, F gives rise to a bijection from the boundary of the disc to the boundary polygon p.

Reference no: EM131350809

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