Intermediate value-like theorem for $\mathbb{C}$? Is there an intermediate value like theorem for $\mathbb{C}$? I know $\mathbb {C}$ isn't ordered, but if we have a function $f:\mathbb{C}\to\mathbb{C}$ that's continuous, what can we conclude about it? 
Also, if  we have a function, $g:\mathbb{C}\to\mathbb{R}$
,continuous with $g(x)>0>g(y)$ does that imply a z "between" them satisfying $g(z)=0$.
Edit:
I apologize if the question is vague and confusing. I really want to ask for which definition of between, (for example maybe the  part of the plane dividing the points), and with relaxed definition on intermediateness for the first part, can we prove any such results?

Consider $f(x) = e^{\pi i x }$. 
We know that $f(0) = 1, f(1) = -1$. 
But for no real value $r$ between 0 and 1 is $f(r) = 0$, or even real valued.
Think about how this is a 'counter-example', and what aspect of $\mathbb{C}$ did we use. It could be useful to trace out this graph.
