Continuity (redirected from continuities)

continuity

In mathematics, a property of functions and their graphs. A continuous function is one whose graph has no breaks, gaps, or jumps. It is defined using the concept of a limit. Specifically, a function is said to be continuous at a value x if the limit of the function exists there and is equal to the function's value at that point. When this condition holds true for all real number values of x in an interval, the result is a graph that can be drawn over that interval without lifting the pencil. Such functions are crucial to the theory of calculus, not just because they model most physical systems but because the theorems that lead to the derivative and the integral assume the continuity of the functions involved.

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continuity Film, TV

1. the comprehensive script or scenario of detail and movement in a film or broadcast

2. the continuous projection of a film, using automatic rewind

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continuity [‚känt·ən′ü·əd·ē]

(civil engineering)

Joining of structural members to each other, such as floors to beams, and beams to beams and to columns, so they bend together and strengthen each other when loaded. Also known as fixity.

(electricity)

Continuous effective contact of all components of an electric circuit to give it high conductance by providing low resistance.

(navigation)

The ability of a navigational system to let the user navigate without interruption.

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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.

REFERENCES

Dedekind, R. Nepreryvnost’ i irratsional’nye chisla, 4th ed. Odessa, 1923. (Translated from German.)
Cantor, G. “Osnovy obshchego ucheniia o mnogoobraziiakh.” (Translated from German.) In Teoriia assemblei, vol. 1. St. Petersburg, 1914. (Novye idei v matematike, collection 6.)
Hilbert, D. Osnovaniia geometrii. Moscow-Leningrad, 1948. (Translated from German.)
Hausdorff, F. Teorüa mnozhestv. Moscow-Leningrad, 1937. (Translated from German.)
Aleksandrov, P. S. Vvedenie v obshchuiu teoriiu mnozhestv i funktsii Moscow-Leningrad, 1948.