Definition of Holomorphic function

Question

If a function $f$ is not holomorphic at $z_0$, then for each $\epsilon>0$, $f$ is not differentiable on the open ball $B_{\epsilon}(z_0)$. Does this imply that $f$ is not differentiable at $z_0$? Is there a function $f$ that is complex differentiable at $z_0$ but not on any open ball around $z_0$?