# Differentiable but not Holomorphic?

Can I get a few examples of complex functions being complex differentiable at a point but not holomorphic in their domain?

• Do you mean real differentiable, or even $C^{\infty}$, but not holomorphic? I think you need to specify what you mean. – Moya Aug 17 '15 at 17:30
• What exactly do you mean? Differentiable in the complex sense at some point? – Robert Israel Aug 17 '15 at 17:30
• I assume you mean differentiable interpreted as a function $\Bbb R^2\to\Bbb R^2$. Take any polynomial in ${\rm Re}(z)$ and ${\rm Im}(z)$ that is not a polynomial in $z$. For example, ${\rm Re}(z)$ and ${\rm Im}(z)$ themselves. If you mean complex-differentiable in a nbhd of a point but not holomorphic at the point, then you'll run into a problem: there aren't any. Not sure what can be said pointwise. – whacka Aug 17 '15 at 17:31

$$f(z) = |z|^2$$ 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.