# Piecewise Affine Bijections of $\mathbb{R}^n$

I have a min-max function $f:\mathbb{R}^n\to\mathbb{R}^n$ of the form $$f(x) = \min_{i=1,\dots,n}\max_{j=1,\dots,n}(\alpha_{ij}^Tx + \beta_{ij})\quad\text{where each } \alpha_{ij}\in \mathbb{R}^{n\times n}\text{ and }\beta_{ij}\in\mathbb{R}^n.$$ I'm looking for a sufficient condition(s) under which $f$ is a bijective function of $x$.

To that end, I've found a paper on Piecewise Affine Bijections of $\mathbb{R}^n$ by Dieter Kuhn and Rainer Löwen 1987, in which there are a few theorems on necessary and sufficient conditions for a piecewise affine function $f$ on $\mathbb{R}^n$ to be bijective. However, these theorems require certain properties of $f$ which are not so easy to verify. One of the theorems states that: if $f$ is proper, then it is bijective if and only if $f$ is injective. So I don't know how we can show that $f$ is injective!

I appreciate any hint or suggestion.

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