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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 ...
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.
The derivative at a non-essential singularity itself has a non-essential singularity, with increased by 1 (except if is 0 so that the singularity is removable). The point a {\displaystyle a} is an essential singularity of f {\displaystyle f} if it is neither a removable singularity nor a pole.
Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory. [ 1 ] [ 2 ] [ 3 ] The functions are notated with brackets, as x − a n {\displaystyle \langle x-a\rangle ^{n}} where n is an integer.
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]
the fact that the radius of convergence is always the distance from the center to the nearest non-removable singularity; if there are no singularities (i.e., if is an entire function), then the radius of convergence is infinite. Strictly speaking, this is not a corollary of the theorem but rather a by-product of the proof.
For example, in the classification of discontinuities: 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);
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