A continuous nowhere differentiable function having additive property.

Question

I'm not sure whether this is duplicated or not.

What I'd like to find is an example of a continuous nowhere complex differentiable function satisfying $f(z_1+z_2)=f(z_1)+f(z_2)$ for all $z_1, z_2$ in $ \mathbb{C}$.

I already know Weierstass function is an example of real valued continuous but nowhere differentiable function.

But is there such a complex valued function satisfying above conditions?

Any reference or direct explanation would be helpful.

Thank you.

Answer 1

It is well-known that every continuous additive function on $\Bbb R$ is linear. In particular, there are $c_1,c_2$ such that $f(t)=c_1t$ and $f(it)=c_2t$. Then $f(x+iy)=c_1x+c_2y$