In complex analysis, if any complex function is analytic and also this function has singular points. Then why this complex function is analytic everywhere except those singular points????
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This depends on what do you mean by a singular point. Usually, we say singular if somewhere nearby the object is regular. For example, if function is analytic in a neighbourhood of a point, but not at this point, then we say that it has an isolated singularity.
On the other hand, there are functions that are not analytic at all. Let us consider f(z)=|z| . This function is not complex differentiable at any point. Probably, you can say that it has a singularity at all points, but I would not use this language.
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