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Is it possible to construct a compact superset based on a closed continuous interval? [closed]

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Let $I = [a,b] \subset \mathbb{R}$ be a closed interval and $D \subset \mathbb{R}^{n}$ be compact. Furthermore, $g : D \to \mathbb{R}^{n \times n}$ is a continuous function and $f: I \times D \to \mathbb{R}^{n \times n}$ is another function with $f(h,x) = h g(x)$. In this case, we know that the image $f(h, D) = hg(D)$ is compact for some $h \in I$. Is there a way to construct a compact set $S \subset \mathbb{R}^{n \times n}$ such that $f(h, D) \subset S$ is satisfied for all $h \in I$?


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