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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.
Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers. Determining whether a given initial point with rational coordinates is periodic, or whether it lies in the basin of attraction of a given open set, in a piecewise-linear iterated map in two dimensions ...
The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation y 2 − x 3 = 0 defines a curve that has a cusp at the origin x = y = 0. One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at ...
One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionals, ordinary and partial differential equations ...
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 computation of the Hausdorff dimension of the graph of the classical Weierstrass function was an open problem until 2018, while it was generally believed that = + <. [ 6 ] [ 7 ] That D is strictly less than 2 follows from the conditions on a {\textstyle a} and b {\textstyle b} from above.
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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