There is a parameter $\alpha\in A \equiv [\underline{\alpha}, \overline{\alpha}]$. There are two functions $f,g:[0,1] \times A \rightarrow\mathbb{R}$ which are continuous. I want to maximize $f(x;\alpha)$ wrt $x$, subject to $f(x;\alpha)=g(x;\alpha)$, and show that the maximized value of $f(x;\alpha)$ is a continuous function of $\alpha$, i.e. a standard Maximum Theorem application.
My problem is such that it can be shown that for each $\alpha \in A$ there exist $x,x' \in [0,1]$ such that $f(x)-g(x)>0$ and $f(x')-g(x')<0$, i.e. for each $\alpha$, $f(x;\alpha)=g(x;\alpha)$ has a solution in $[0,1]$, i.e. the feasible set is non-empty. Let the feasible set be $X(\alpha)$, i.e. $X(\alpha)=\{x\in[0,1]:f(x;\alpha)=g(x;\alpha)\}$. The part I'm struggling with is - how to show the correspondence $X$ is upper and lower hemicontinuous.
I think each particular solution to $f(x;\alpha)=g(x;\alpha)$ does change continuously with $\alpha$, due to continuity of $f$ and $g$, but not sure how to extend this to uhc and lhc.