= Complex differentiability at a point without holomorphy nearby
{title2=$f(z)=z/(1+|z|)$}
At zero, $[f(h)-f(0)]/h\to1$, so the <complex derivative> exists. At a nonzero point, write $r=|z|$. The <Cauchy-Riemann equations> for $x/(1+r)$ and $y/(1+r)$ force $x^2=y^2$ and $xy=0$, impossible when $r>0$. Hence the function is <complex differentiable> only at zero and is <holomorphic> on no nonempty open subset. A <complex derivative> at one point gives no neighborhood of <complex differentiability>.
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