Interest in the study of the properties of functions defined on the set of \textit{h}-complex numbers arose again in connection with existing applications in geometry and mechanics. In this paper, we present necessary and sufficient conditions for \textit{h}-differentiability and \textit{h}-holomorphy of functions of an \textit{h}-complex variable, the theorem on finite increments is proved, sufficient conditions for \textit{h}-analyticity are found, a uniqueness theorem for \textit{h}-analytic functions is proved.