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Satisfying the Cauchy-Riemann equations at a point is, on its own,

  • A.irrelevant to analyticity, which depends only on continuity
  • B.not sufficient for analyticity without continuity of the partial derivatives
  • C.both necessary and sufficient for analyticity at that point
  • D.sufficient for analyticity but not necessary

Answer: B

The equations are necessary, and they become sufficient once the four partial derivatives are also continuous at the point. Without that extra hypothesis there are functions satisfying them at the origin yet not differentiable there, so the equations alone cannot be treated as a complete test

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