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Ill a lecture presented by G. T. Whyburn' at the 1955 Summer Institute on Set Theoretic Topology, attention was called to the lack of a purely topological proof of the fact that a mapping generated by an analytic function f of a complex variable can be a local homeomorphism at a point zO of the complex plane only if f'(zo) # 0. It is the purpose of this note to give such a proof of this fact. The notation of an earlier paper2 will be employed, and the proof will constitute a variation of a proof contained in that article. No distinction in notation will be employed to distinguish the idea of a "function of a complex variable" from the idea of "the mapping generated by a function of a complex variable." THEOREM. Let f(z) be a function of the complex variable z, analytic in the region E of the complex plane, and suppose that f is a local homeomorphism at zO e E. Then f'(zO) # 0. Proof: Suppose that f is a local homeomorphism at zo, but f'(zo) = 0. We shall employ Z and W to represent the original and the image planes, respectively. Let D be a region containing zo and contained in E such that f maps D topologically ontof(D) c W. Let A = R + C be a disk centered at zo, with circular boundary C and interior R and such that A c D. Let ¢: [a, b] -e C be a positive traversal of C.' Then, since f is a homeomorphism on A, fi is a traversal of the simple closed curve f(C) and wo = f(zo) is a point interior to f(C). Consequently,3 gc(f, wo) = /'ab(ft, wo) = +27ri. Definev: Z -W by the equations

A TOPOLOGICAL PROOF OF A THEOREM OF COMPLEX ANALYSIS