Show that $f(z) = \mathrm{Re}(z)\mathrm{Im}(z)$ is complex differentiable only on $0$. Suppose we need to find $f'(z\_0)$. When $z\_0 = 0$ then $$\lim\_{\Delta z \to 0} {\dfrac{f(0+\Delta z) - f(0)}{\Delta z} = \lim\_{\Delta z \to 0} {\dfrac {\mathrm{Re}(\Delta z)\mathrm{Im}(\Delta z)}{\mathrm{Re}(\Delta z)+i\mathrm{Im}(\Delta z)}}} = \lim\_{\Delta z \to 0}{\dfrac{1}{1/\mathrm{Im}(\Delta z) + i/\mathrm{Re}(\Delta z)}} = 1/\infty = 0$$

But I can't show that the complex derivative doesn't exist on any other $z\_0$. Any hints? Thanks in advance.