Consider the parametric linear problem
$ x^*(\theta) := \min_{Y , \ Z } \left\| Z \right\|_1 $
$ \text{sub. to: } \ \theta A + B Y = \theta C Z.$
where $Y \in \mathbb{R}^{m \times s} $, $Z \in \mathbb{Z} := \{M \in (\mathbb{R}_{\geq 0})^{s \times s} \mid \left\| M \right\|_1 \leq 1 \}$; $\ A \in \mathbb{R}^{n \times s}$, $B \in \mathbb{R}^{n \times m}$, $C \in \mathbb{R}^{n \times s}$ are given non-null matrices; $\underline{1}^\top = (1,1,...,1) \in \mathbb{R}^s$.
Let $\theta \in [\epsilon,1]$ for some $\epsilon>0$.
Is the function $\theta \mapsto x^*(\theta)$ continuous?
Edit: Say what happens if $Y \in \mathbb{Y}$, being $\mathbb{Y} \subset \mathbb{R}^{m \times s}$ a compact set.
Note: this question is simpler than this one.