Branch cuts imaginary part This is a very simple gap in my intuition about branch cuts. I have heard informally a statement of the type: if a complex function suddenly acquires an imaginary part at the point $x_0$ in the real axis, this point is a branch point.
Is this necessarily true? What is the scope of this affirmation and why is it true? Also, why are the functions that are multivalued (with respect to the polar coordinate on the complex plane), always multivalued only on its imaginary part? Why isn't there functions whose real part is also multivalued?
I'd appreciate a formal answer, but any insight is greatly appreciated.
 A: I think there is something garbled in the statement...
First, it is not clear to me what "suddenly" means. :)  The imaginary part of a holomorphic function may or may not "suddenly" become non-zero, depending on what one means by this.
But/and in any case, this is not a great criterion for "a branch point"... Possibly/presumably, $f(z)=\sqrt{z}$ is a prototype for the phenomenon dubiously described as above, in the sense that, yes, "suddenly", the square root becomes non-real as $z$ moves across $0$. But/and, yes, this does also presume that we have some definition/description of $\sqrt{z}$ that somehow extends "across" $0$... which is already an issue.
Altogether, I'm not a fan of whatever a corrected/clarified version of this "criterion" turns out to be, because (to my perception) it masks the genuine phenomena.
(A more genuine point is that "branch point" is an intrinsic thing, but "branch cut" is definitely not.)
EDIT: just to have an example: $f(z)={\root 3 \of z}$ can be construed as being real both on $z$ real and positive, and $z$ real and negative. But/and there is a branch point at $z=0$.
