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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 . [a] This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's ...
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. [8] [9]
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to ...
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.
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.
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Then, the point x 0 = 1 is a jump discontinuity. In this case, a single limit does not exist because the one-sided limits, L − and L +, exist and are finite, but are not equal: since, L − ≠ L +, the limit L does not exist. Then, x 0 is called a jump discontinuity, step discontinuity, or discontinuity of the first kind.
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.