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jostpuur
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Is it possible for a complex analytic function to have an uncountable set of singular points?
HallsofIvy said:Analytic where? Obviously, a function that is analytic everywhere has NO singular points. Am I correct that by "singular point" you mean a point at which the function is not analytic?
Certainly it would be possible to define a function that would be analytic everywhere except at certain points and I see no reason why one could not do that for and uncountable set of points. The only requirement would be that the set of points on which the function is not analytic would be a closed set.
mathwonk said:it depends what kind of functions you want to allow. you want of course a function which is analytic on some open, presumably connected set in C, and which cannot be analytically continued outside that set to another strictly larger such open connected set, right?
A singular point, also known as a singularity, is a point in a mathematical function or surface where the function or surface is not well-defined or differentiable. This means that at a singular point, the function or surface is either undefined or has a discontinuity in its derivative.
The amount of singular points can be determined by taking the derivative of the function or surface and finding the points where the derivative is equal to zero or undefined. These points are known as critical points and may or may not be singular points.
Singular points can provide important information about the behavior of a function or surface. They can indicate where the function is not smooth or where there may be a change in the behavior of the function. In some cases, singular points can also help identify critical points that may be local maxima or minima.
Yes, it is possible for a function or surface to have an infinite amount of singular points. An example of this is the function f(x) = 1/x, which has a singular point at x=0 and an infinite amount of singular points as x approaches 0.
Singular points can be classified as either removable or non-removable. Removable singular points can be "smoothed out" by redefining the function at that point, while non-removable singular points cannot be smoothed out and must be treated separately. Singular points can also be classified as isolated or non-isolated, depending on whether they occur in isolation or as part of a larger set of singular points.