continuity


continuity
/kon'tn ooh"i tee, -tn yooh"/, n., pl. continuities.
1. the state or quality of being continuous.
2. a continuous or connected whole.
3. a motion-picture scenario giving the complete action, scenes, etc., in detail and in the order in which they are to be shown on the screen.
4. the spoken part of a radio or television script that serves as introductory or transitional material on a nondramatic program.
5. Math. the property of a continuous function.
6. Usually, continuities. sets of merchandise, as dinnerware or encyclopedias, given free or sold cheaply by a store to shoppers as a sales promotion.
[1375-1425; late ME continuite < AF < L continuitas, equiv. to continu(us) CONTINUOUS + -itas -ITY]
Syn. 2. flow, progression.

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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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      in mathematics, rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable—say x—is associated with a value of a dependent variable—say y. Continuity of a function is sometimes expressed by saying that if the x-values are close together, then the y-values of the function will also be close. But if the question “How close?” is asked, difficulties arise.

      For close x-values, the distance between the y-values can be large even if the function has no sudden jumps. For example, if y = 1,000x, then two values of x that differ by 0.01 will have corresponding y-values differing by 10. On the other hand, for any point x, points can be selected close enough to it so that the y-values of this function will be as close as desired, simply by choosing the x-values to be closer than 0.001 times the desired closeness of the y-values. Thus, continuity is defined precisely by saying that a function f(x) is continuous at a point x0 if, for any degree of closeness ε desired for the y-values, there is a distance δ for the x-values (in the above example equal to 0.001ε) such that for any x within the distance δ from x0, f(x) will be within the distance ε from f(x0). In contrast, the function that equals 0 for x less than or equal to 1 and that equals 2 for x larger than 1 is not continuous at the point x = 1, because the difference between the value of the function at 1 and at any point ever so slightly greater than 1 is never less than 2.

      A function is said to be continuous on an interval if it is continuous at each point of the interval. The sum, difference, and product of continuous functions are also continuous, as is the quotient, except at points at which the denominator is zero. Continuity can also be defined in terms of limits (limit) by saying that f(x) is continuous at x0 if

      A more abstract definition of continuity can be given in terms of sets, as is done in topology, by saying that for any open set of y-values, the corresponding set of x-values is also open. (A set is “open” if each of its elements has a “neighbourhood,” or region enclosing it, that lies entirely within the set.) Continuous functions are the most basic and widely studied class of functions in mathematical analysis, as well as the most commonly occurring ones in physical situations.

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Universalium. 2010.

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