The complex derivative exists when this limit has the same finite value for increments approaching zero from every complex direction. For , real and imaginary increments give the necessary Cauchy-Riemann equations. Existence of those partial derivatives at one point alone is not sufficient for complex differentiability at a point; continuous partial derivatives in a neighbourhood, together with the Cauchy-Riemann equations there, give a standard sufficient condition for to be a holomorphic function.
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