\textit{D. Menchoff} [Math. Ann. 95, 640-670 (1926)] established that if a univalent continuous function f: \(D\to {\mathbb{C}}\) of a complex variable satisfies a certain condition K' in each point of D except possible in a countable set, then f is holomorphic in D. \textit{Yu. Yu. Trokhimchuk} [``Continuous mappings and the conditions of monogeneity'' (Russian), 212 p. (1963; Zbl 0133.038)] showed that this theorem still holds even without the hypothesis of the injectivity. The author generalizes Menchoff theorem in the case of continuous mappings f: \(D\to {\mathbb{C}}^ n\) of n complex variables and which are not supposed to be univalent.