Let $p(x_1,x_2)=x_2^n+\text{terms with degree less than }n$ with real coefficients.
There exist continuous functions $z_1(x_1),...,z_n(x_1)$ from $[0,1]$ to $\mathbb{C}$ such that $p(x_1,z_i(x_1))=0$ for every $i$.
It is possible for $p$ to have a root with multiplicity greater 1 for every $x_1\in[0,1]$ such as $(x_1-x_2)^2$.
It's also possible for the multiplicity of a root to change like in $x_2^2-x_1$ (the multiplicity of $z_1(x_1)=-\\sqrt{x_1}$ is $1$ for $x_1\ne 0$ and $2$ for $x_1=0$)
My question is the following: does the multiplicity of the root $x_2=z_1(x_1)$ (the number of $i$ such that $z_1(x_1)=z_i(x_1)$) change finitely many times for $x_1\in [0,1]$? If the number of changes is finite, can the number be bounded in terms of $n$?
By looking at a graph I know that the answer for $p(x_1,x_2)=p_1(x_2)+x_1$ is yes for the real roots (I'm not sure about the complex roots though) and the bound for the real roots is $n-1$ (the number of times $p_1'(x)=0$)