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A function of a complex variable is called analytic at a point when it is differentiable

  • A.throughout some neighbourhood of the point
  • B.at the point itself and nowhere else needed
  • C.along at least one direction through the point
  • D.at every point of the complex plane

Answer: A

Complex differentiability at an isolated point is too weak to be useful; analyticity demands it on a whole open disc about the point. That extra requirement is what buys the powerful results -- infinite differentiability, a convergent Taylor series, and Cauchy's theorem -- none of which follows from differentiability at one point alone

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