# Discontinuous Function

Also found in: Dictionary, Wikipedia.

## 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 ?
The partition of unity enrichment with discontinuous functions allows to model discontinuity in the domain such as a crack in an elastic medium.
Sukumar, "Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons," Computational Mechanics, vol.
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.
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.
A fast algorithm for evaluating the Fourier transform of 2D discontinuous functions has been proposed in  as an extension of  to allow an arbitrary boundary shape.
Arbitrarily, I have made the function cross the 45-degree line, to highlight the fact that discontinuous functions can still have a point where f(x) = x.
f (x, t) = w(x)z(t), and compute [f.sup.h,[tau]] = [w.sup.h][z.sup.[tau]] accordingly to the type of w and z (i.e., distributions or discontinuous functions).
The change in the parameters of the electric circuit is described by means of discontinuous functions and is found in the differential equation for the sought value.
Keywords: Discontinuous Functions Interpolation Approximation Functional Spaces Fractals.
Adaptive ENO-wavelet transforms for discontinuous functions, Technical Report 99-21
The modern city of doing is one of discrete and discontinuous functions dispersed in different locations (home, workplace, sports field) requiring different modes of behaviour (parent, employee, athlete or fan) dispersed in a spatial and experiential void.
(2c,d) abruptly truncate the incident field at the edge and, thus, transform these integrands into discontinuous functions that violate the wave equation.

Site: Follow: Share:
Open / Close