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What are some examples of functions that are differentiable (everywhere) in $\mathbb{R^2}$, but that are not differentiable in the complex plane? We got an example for homework, $f(z)=2xy$, and I was wondering if there were any others.

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    Almost any $\mathbb R^2 \to \mathbb R^2$ function you come up with randomly will fail to be complex differentiable. Complex differentiability is a pretty stringent condition.2012-02-10
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    Your $f(x,y)=(2xy,0)$ is a special case of the fact that $f(x,y)=(u(x,y),0)$ is never complex differentiable unless $f$ is constant, which can be seen as a special case of the Cauchy-Riemann equations or the open mapping theorem.2012-02-10
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    Related: [How is $\mathbb{C}$ different than $\mathbb{R}^2$?](http://math.stackexchange.com/q/5108)2012-02-10

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