Is following contradictory?
"Then $f = u + iv$ is complex-differentiable at that point if and only if the partial derivatives of $u$ and $v$ satisfy the Cauchy–Riemann equations (1a) and (1b) at that point. The sole existence of partial derivatives satisfying the Cauchy–Riemann equations is not enough to ensure complex differentiability at that point." - 1[1, second paragraph]
They say that $f$ is complex-differentiable iff partial derivatives of $u$ and $v$ satisfy C-R equations, but still it is not enought to ensure complex differentiability at that point.
So do you need extra conditions as wikipedia says or not for $f$ to be complex differentiability? Can you give me example, where function satisfy C-R equations, but is not Complex differentiable at certain point?