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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. – Rahul Feb 10 '12 at 5:58
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. – Jonas Meyer Feb 10 '12 at 6:01

As some commentors have pointed out, there are many, many other such functions. All that's required is that your function fail to satisfy the Cauchy-Riemann equations -- that is, if $f(z) = u(z) + iv(z)$, the function $f$ will fail to be complex differentiable just in case one of the following fails:

$u_x(z) = v_y(z)$

$u_y(z) = -v_x(z)$.

Here are a few classic examples that are easily seen to be real-differentiable:

$f(z) = \overline z$

$f(z) = |z|$

EDIT: although the latter fails to be real-differentiable on the axes (my mistake).

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I don't think your second example is $\mathbb{R}$-differentiable everywhere... – Najib Idrissi Feb 10 '12 at 6:43
Ah, you're quite right. My mistake. I'll edit that now. – samhop Feb 12 '12 at 22:23
$|z|$ is real differentiable at most points on the axes. The only point where it is not differentiable is the origin. – Jonas Meyer Feb 12 '12 at 23:10

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