The author considers functional equations of form (1). It is asserted that the process of solving (1) is determined by the kernel $\varphi(x)$. If the kernel consists of stationary points only, the solution of equation (1) is reduced to the solution of a system of ordinary equations; if the kernel has no stationary points, the method of steps is employed.
If the kernel has no more than an even number of stationary points, then (see [4]) the real axis is divided into sets $M-i,\; i = 0, 1, 2, ...$ so that $\varphi(M_i) \subseteq M_i$. The method of steps is applied to each set except $M_0$.
The idea of dividing the region of definition of the function into invariants in respect to the function of sets is then used once more for determining the nature of the solution obtained.

Citation Example:Sharkovsky O. M. On the solution of a class of functional equations // Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 86-94.