How to show a correspondence is lower-hemicontinous?

Question

I have a composite function $f:X \rightarrow Y$, such that $X\subset\mathbb{R}^n,Y\subset\mathbb{R}^{k\times k}$, where $f$ involves a combination of transformations:

1. Form a finite number $p$ of matrix polynomials in $x_1,x_2,\cdots,x_n$. Call them $A_1,\dots,A_p$.
2. The matrices $A_1,\dots,A_p$ form another polynomial in $z\in \mathbb{C}$, i.e., $A_1 z^{-q}+A_2 z^{-q+1}+\cdots+A_p z^r=0$
3. Apply QZ decomposition to $A_1$ (or simply a Wiener Hopf factorization of the whole polynomial)
4. Perform $[\cdot]_+$ to one of the factorized matrices to get rid of the negative order terms in $z$.
5. Perform matrix addition, multiplication and inversion of the resulting matrices.

Now I would like to show (if possible) $f^{-1}$ is in fact a continuous correspondence at a generic point, or almost everywhere.

Since I have a bounded space, and $f$ is continuous (can show it is analytic bc Wiener-Hopf preserves analyticity), proving upper-hemicontinuity should not be a problem.

My question is, how do I proceed to show its lower-hemicontinuity at a certain point? If not, what conditions are needed?

It looks like if $f$ is defined on complex domain, by open mapping theorem and the fact any open mapping $f$ its inverse correspondence is lower-hemicontinuous we are done here. But the problem is $X$ is a Euclidean space. Not sure if this is ok.

Thank you!

Here's a definition of lower- hemicontinuity from Efe A.Ok Real Analysis with Economic Applications (2007)

Proposition 4. Let $X$ and $Y$ be two metric spaces, and $\Gamma: X \rightrightarrows Y$ a correspondence. $\Gamma$ is lower hemicontinuous at $x \in X$ if, and only if, for any $\left(x^{m}\right) \in X^{\infty}$ with $x^{m} \rightarrow x$ and any $y \in \Gamma(x)$, there exists a $\left(y^{m}\right) \in Y^{\infty}$ such that $y^{m} \rightarrow y$ and $y^{m} \in \Gamma\left(x^{m}\right)$ for each $m$.

Answer 1

Alternative definition of upper hemicontinuous: if $ f ^{-1}$ has a closed graph and the images of compact sets in $Y$ are bounded in $X$, $ f ^{-1}$ is UHC (MasCollel, Whinston and Green (1995), Appendix M.H.).

Now, given that you have a finite number of matrix polynomials of some degree (OP didn't specify), boundedness of compact sets should not be an issue.

For the closed graph condition, check all points in $\{ (y,x) \in Y \times X : x \in f^{-1}(y)\}$.

You can do this for all non-measure zero sets to see if those two conditions hold. Seems non-trivial in your case but maybe you can work out the details from here I hope :)