Discontinuous Function

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Discontinuous Function

a function that is discontinuous at some points. In the functions usually encountered in mathematics, points of discontinuity are isolated, but there exist functions that are discontinuous at all points. An example is the Dirichlet function: f(x) = 0 if x is rational and f(x) = 1 if x is irrational.

The limit of a sequence of continuous functions that converges everywhere may be a discontinuous function. Such discontinuous functions are called functions of the first Baire class, after the French mathematician R. Baire, who provided a classification of discontinuous functions. Measurable discontinuous functions are an important class of discontinuous functions.

H. Lebesgue constructed a theory of the integration of discontinuous functions. N. N. Luzin showed that by changing the values of a measurable function on a set of arbitrarily small measure the function can be made continuous. If a function is monotonic, then it has only jump discontinuities. For functions of several variables, not only isolated points of discontinuity but also, for example, lines and surfaces of discontinuity must be considered.

REFERENCE

Baire, R. Teoriia razryvnykh funktsii. Moscow-Leningrad, 1932. (Translated from French.)
References in periodicals archive ?
Given the locations of discontinuities, the essence of the IPRM is solving an inverse problem from the already-known finite Fourier series of a discontinuous function and that of the polynomial basis functions.
In this section, first the fast IPRM will be presented for reconstructing a single-region discontinuous function;, and then it will be extended to a multiple-region discontinuous function.
The Fast IPRM Method for a Single-region Discontinuous Function
As a discontinuous function can be divided into multiple regions of piecewise continuous functions, we first consider the fast IPRM method for only one region of such a function.
It is now shown that the sort of mechanism modeled by Stein requires the public's expectations to be a discontinuous function of government announcements.
Stein's solution does remain a possibility, though his solution can only be maintained if public expectations are a discontinuous function of announcements.
If we assume the median voter hypothesis, then the median Congressional reaction may be a discontinuous function of announcements, in accordance with the equilibrium, even if Congressional reactions to announcements are not unanimous; all that is necessary is that the median Congress-person act in accordance with the equilibrium (see Conlon |3~).
The previous section showed that cheap talk equilibria of the type developed by Stein |22~ depend upon public reactions which are discontinuous functions of government announcements.
A fast algorithm for evaluating the Fourier transform of 2D discontinuous functions has been proposed in [12] as an extension of [11] to allow an arbitrary boundary shape.
The proposed algorithm is very useful for many applications, where the Fourier transform of discontinuous functions in an arbitrary shape boundary area is required to be computed, such as in antennas, scattering, computational electromagnetics, signal and image processing.
Liu, "DIFFT: a fast and accurate algorithm for Fourier transform integrals of discontinuous functions," IEEE Microwave and Wireless Components Letters, Vol.
473) an illustration of that supposedly rarest of flowers, a discontinuous function.

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