Is function $f(z)=|z-9|$ differentiable?
Solution 1:
There is no need to use the Cauchy-Riemann equations here. The function $f$ is not differentiable at $9$ because the limit$$\lim_{z\to9}\frac{f(z)-f(9)}{z-9}=\lim_{z\to9}\frac{|z-9|}{z-9}$$doesn't exist. It is equal to $1$ if $z\in 9+\Bbb R$ and it is equal to $-i$ if $z\in9+i\Bbb R$.