This is a literature request for (hopefully) an English version to a rigorous proof that a complex algebraic curve cannot abruptly end.

That is, if the algebraic curve enters a closed region it must also leave it.

This has a historic significance because Gauss's proof in his Phd thesis assumed this property holds. From looking around it seems that A Ostrowski rigorously proved the result around the 1920's. Is this correct? I am unable to find the title of the paper.

Is there also a proof that a real algebraic curve does not end abruptly?

I don't regard this property as obvious, but it doesn't seem to be well commented in the literature. Maybe, I'm wrong.

Thanks in advance.

Abruptly end: Given an irreducible polynomial $p$, we define $V(p)$ to be the complex algebraic curve associated to $p$. I say that $V(p)$ does not abruptly end at $(x,y)\in\mathbb{C}^2$ with $p(x,y)=0$ if there is a disc small enough so that the boundary contains exactly two points in $V(p)$.

(This is a first attempt and maybe needs some corrections.)