TITLE: Differentiable but not Holomorphic?
QUESTION [5 upvotes]: Can I get a few examples of complex functions being complex differentiable at a point but not holomorphic in their domain?

REPLY [6 votes]: $$
f(z) = |z|^2
$$
This is differentiable at just one point, $z=0$.  If "holomorophic at $0$" were construed as meaning differentiable everywhere in some open neighborhood of $0$, then this would not be holomorphic at $0$.  But I'm not sure that conventional definitions are sufficiently standard that you can say "holomorphic but not differentiable" and have everybody understand it in the same way.