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Say that a matrix valued function $A: \mathbb{R} \rightarrow \mathbb{R}^{n \times n}$ is upper semi-continuous at $x_0$ if

$$ \limsup_{x \rightarrow x_0} A(x) \preceq A(x_0), $$

where $\preceq$ denotes the Loewner order; and that $A$ is upper semi-continuous if it is upper semi-continuous at all points $x_0 \in \mathbb{R}$.

Moreover, say that $A: \mathbb{R} \rightarrow \mathbb{R}^{n \times n}$ is non-increasing if for any $s,t \in \mathbb{R}$, $$ s \le t \implies A(s) \preceq A(t) .$$

It is known that for $n=1$, the composition $f \circ g$ of an upper semi-continuous function $f:\mathbb{R}^{n \times n}\rightarrow\mathbb{R}$ and an upper semi-continuous and non-increasing $g:\mathbb{R} \rightarrow\mathbb{R}^{n \times n}$ is upper semi-continuous.

Is this fact also valid for $n>1$?

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    $\begingroup$ It seems to me that the composition isn't defined when $n > 1$, since both functions have domain $\mathbb R^n$ and codomain $\mathbb R^{n \times n}$. Am I misunderstanding, or did you mean something else? $\endgroup$ Sep 19, 2015 at 11:41
  • $\begingroup$ @AaronTikuisis You are right, I forgot to write that the domain of $g$ is $\mathbb{R}$. Thanks for noticing. $\endgroup$
    – Tadashi
    Sep 19, 2015 at 17:02

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