Showing the complex function is differentiable in a set of points

Question

I´ve to show if the function:

$f(x+iy)=2x^2+4iy$

Is differentiable at some points in $\mathbb{C}$.

Checking the Cuchy Riemann equations, the function can only be differentiable at points of the form

$z=1+iy$

Then, taking the limit

$\lim_{h_1,h_2\to0}\frac{f(1+iy+h_1+ih_2)-f(1+iy)}{h_1+ih_2}=\lim_{h_1,h_2\to0}\frac{2h_1^2}{h_1+ih_2}+4=4$

Is this enough to show that the function $f$ is only differentiable at points of the form $z=1+iy$ ?