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.
References in classic literature ?
The nature and relative proportions of the inhabitants of oceanic islands likewise seem to me opposed to the belief of their former continuity with continents.
I had been used to the pleasures of variety, to the luxury and stir of life in Paris; it was only when I had overcome my first repugnance that I saw the advantages of this existence; how it lent itself to continuity of thought and to involuntary meditation; how a life in which the heart has undisturbed sway seems to widen and grow vast as the sea.
He slept through the early gray of morning until the direct rays of the sun smote his closed eyelids, when he awoke with a start and looked about him until he had established the continuity of his existence and identified his present self with the days previously lived.
But the hypothesis of continuity in mental development is clearly preferable if no psychological facts make it impossible.
The hypothesis of mental continuity throughout organic evolution may be used in two different ways.
Yes, time will pass; time, which arranges all things, and the old relations will be reestablished," Alexey Alexandrovitch told himself; "so far reestablished, that is, that I shall not be sensible of a break in the continuity of my life.
Michaelis by staring unwinkingly at the fire had regained that sentiment of isolation necessary for the continuity of his thought.
It is the privi- lege of early youth to live in advance of its days in all the beautiful continuity of hope which knows no pauses and no introspection.
Many other prisoners had had experience of this condition before him, and its violence and continuity had worn themselves out in their cases, as they did in his.
MANAMA: Batelco has won the Best BCI Middle East Award (Best Continuity and Resilience Team 2015).
By attaining Gold Level membership, Oracle has recognized Continuity Software for its commitment to establish Oracle-related knowledge in delivering solutions that mitigate downtime and data-loss risks across the entire enterprise IT environment - including disaster recovery (DR), high availability (HA) and cloud environments; and for uniquely addressing the challenges of joint customers.
Mashreq, one of the UAE's leading financial institutions, recently hosted a Business Continuity "Meet the Experts" (MTE) session.