Abstract
For the first time, the theory of functions of four-dimensional numbers with commutative product was described in works of Abenov M.M., in which the mathematical apparatus was defined, algebraic operations and their properties were determined, functions of four-dimensional numbers, their limits, continuity and differentiability were found. The continuation was the joint work of Abenov M.M. and Gabbasov M.B., where similar anisotropic four-dimensional spaces (withthe notation M2-M7) were defined, which are also commutative with zero divisors. This work is devoted to the study of functions of a four-dimensional variable, definitions and analysis of fourdimensional functions, their properties, as well as the regularity of functions. The purpose of this work is to analyze the definition of functions of four-dimensional variables of the space M5, as well as theorems on the continuity and existence of differentiability of functions of four-dimensionalvariables. This work is descriptive for comparing the spaces of four-dimensional numbers M5 and M3. In the article, theorems on the continuity and differentiability of functions of four-dimensional variables and their properties are proved, and the Cauchy-Riemann conditions are found. The form of trigonometric, exponential, logarithmic, exponential and power functions of four-dimensional variables is determined and the regularity of functions of M5 space is proved. Key words: four-dimensional function, continuity, differentiability, regular function, Cauchy-Riemann condition.