In this paper the continuity of the derivative of an analytic function of a complex variable is proved in an elementary, or purely topological, fashion. Tha t is, no use whatever is made of complex integration or equivalent tools. The desirability of such a proof has been emphasized in Complex analysis by L. V. Ahlfors [l, p. 82], and even more recently in Topological analysis by G. T. Whyburn [2, p. 89], Our proof has been made accessible only by the extensive modern development of the subject of topological analysis (see [2] for rationale and bibliography). The author wishes to express his appreciation to Professor G. T. Whyburn for suggesting the feasibility of attacking this problem at this time. Throughout, we shall be concerned with a nonconstant complex valued function f(z) defined and having a finite derivative at each point of an open connected set E of the complex plane. We shall employ Theorems A and B in the proof of the main theorem.