I am interested in an expression of the form $$x^*=\inf\{\,x\in X:P(x,y)>0\,\}$$ where $P$ is continuous in both $x$ and $y$. With this equation, it would seem that every $y$ is mapped to a value $x^*$ such that I can define a function $x^*(\,y\,).$ Is it generally true, or are there any sufficient conditions, that would allow me to conclude that $x^*$ as "defined" above is continuous in $y$?
I have to admit that I am not quite sure how to think about this.
Thank you very much.