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The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.
An infinite discontinuity is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its graph has a vertical asymptote.
The non-deleted limit of f, as x approaches p, is L if For every ... Since the value at f(0) is a removable discontinuity, ...
The term removable discontinuity is sometimes broadened to include a removable singularity, in which the limits in both directions exist and are equal, while the function is undefined at the point This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's domain.
Then f is a non-decreasing function on [a, b], which is continuous except for jump discontinuities at x n for n ≥ 1. In the case of finitely many jump discontinuities, f is a step function . The examples above are generalised step functions; they are very special cases of what are called jump functions or saltus-functions.
A graph of a parabola with a removable singularity at x = 2 In complex analysis , a removable singularity of a holomorphic function is a point at which the function is undefined , but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.
Singularity functions are a class of discontinuous functions that contain singularities, i.e., they are discontinuous at their singular points.Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory.
in a removable discontinuity, the distance that the value of the function is off by is the oscillation; in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides); in an essential discontinuity, oscillation measures the failure of a limit to exist.