continuity


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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

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, . . . , xm, . . . of elements of X to an element ξ implies the convergence of the elements’ images f(x1), f(x2), . . ., f(xn... 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.

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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