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 $n>1$?