continuity

one of the most important mathematical concepts, encountered in two basic formulations—the continuity of a set and the continuity of a mapping. From a logical point of view, the concept of the continuity of a set precedes that of a function. Nevertheless, historically the concept of a continuous mapping, or continuous function, had undergone mathematical elaboration before the concept of continuity of a set.

The concept of a continuous real function is generalized to arbitrary mappings in the following way. A single-valued mapping y = f(x) of some set X of elements x into a set Y of elements y is said to be continuous if the convergence of a sequence x₁, x₂, . . . , x_m, . . . of elements of X to an element ξ implies the convergence of the elements’ images f(x₁), f(x₂), . . ., f(x_n... to the image f(x) of the limit element x. Thus, the definition of the continuity of a mapping is dependent on limit relations (in our case, the convergence of sequences) being defined on the sets X and Y. In modern mathematics, a set of elements with definite limit relations among them is called a topological space. Concepts characterizing continuity properties of different sets of mathematical objects are now usually set forth in terms of the theory of topological spaces.