On the general solution of the triangle mean value equation

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The purpose of this paper is to prove that for a general type of functional equation every solution which is measurable on a set of positive measure or satisfies the Baire condition on a Baire set of the second category is continuous. The results solve the problems of W. Sander raised in his paper "Verallgemeinerte CauchyFunktionalgleichungen", Aequationes Math. 18 (1978), 357-369. A typical, but not the most general, result is the following.

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