<s>[INST] Is $1/z$ differentiable on $\mathbb{C}\setminus\{0\}$? The way I usually solve these kind of questions is by using the Cauchy Riemann Equations...
$h(z)=\dfrac{1}{z}$
$=\dfrac{1}{x+iy}$
$=\dfrac{1}{x+iy} \dfrac{x-iy}{x-iy}$
$=\dfrac{x-iy}{(x+iy)(x-iy)}$
$=\dfrac{x-iy}{x^2+y^2}$
$=\dfrac{x}{x^2+y^2}-\dfrac{iy}{x^2+y^2}$
Does this make $u(x,y)=\dfrac{x}{x^2+y^2}$ and $v(x,y)=-\dfrac{y}{x^2+y^2}$
Im a little confused with the example.. or is there a better way for me to show whether it is differentiable or not?
 [/INST] Hint:
$$\frac{1}{x+iy}\not=\frac{1}{x}+\frac{1}{iy}.$$ 
However, 
$$
\frac{1}{x+iy}=\frac{1}{x+iy}\frac{x-iy}{x-iy}
$$
Does this help?
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