Hello.

I have a research note coming out soon, and I'm stuck showing that a weird kind of function is continuous. I need to it show a new method of bounding exponential growth factors in combinatorial classes.

The function in question is $f: [0,1] \rightarrow (0,\infty)$, which sends a parameter $l \in [0,1]$ to the $z$ value of the unique positive critical point ($P''>0$) of a function $P_{S,l}(z) = \sum_{(i,j) \in S} z^{j\cdot l + i\cdot(1-l)}$, where $S \subset \{ 0,1,-1 \}^2$.

For several different sets $S$, I have numerical experiments supporting the claim that this function is continuous. I've considered trying the $L^2$ norm, but I don't get very far before I'm swamped with unmanageable amounts of output. I'm looking for a shortcut that will give continuity, I'm not really concerned with how refined the bounds are. Any help or references are greatly appreciated.

Cheers, Sam