Complex differentiability at a point

A function is complex differentiable at $p$ whenIf $f^{\prime }(p)\ne 0$, the real derivative is multiplication by a nonzero complex number, hence a rotation and scaling; this is the local source of angle preservation by holomorphic maps.

The complex derivative exists when this limit has the same finite value for increments $h$ approaching zero from every complex direction. For $f=u+iv$, 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 $f$ to be a holomorphic function.