Show that $f(z) = \text{Im }z$ is not differentiable I want to prove that $f(z) = \text{Im }z$ is not differentiable anywhere. I know how to prove it easily with the Cauchy-Riemann equations, however I'm also interested in proving it by just using the definition of differentiability. 
I know that the definition of differentiability for complex functions is almost the same as for real functions, but I'm still having problem proving it since we're only considering the imaginary part. 
Thanks in advance

For any $z\in \mathbb{C}$ consider the limit of the difference quotient in two different directions, namely in the purely imaginary direction and also in the real direction. These limits will not agree, so $\text{Im }z$ is not complex differentiable.
