Complex differentiable

Definition

A function may be complex differentiable at a point but not holomorphic at this point. For example, the function ⁠${\displaystyle \textstyle f(z)=\vert z\vert {\vphantom {l}}^{2}=z{\bar {z}}}$⁠is complex differentiable at ⁠${\displaystyle 0}$⁠, but is not complex differentiable anywhere else, especially including in no place close to ⁠${\displaystyle 0}$⁠ (see the Cauchy–Riemann equations, below). So, it is not holomorphic at ⁠${\displaystyle 0}$⁠.