Let $f:\mathbb R^{m\times n}\to\mathbb R^m$ be a continuous function.
$x\in\mathbb R^{m\times n}$.
$g:\mathbb R^{m\times n}\to \mathbb R^n$.
$g(x)$ is a column vector, and $f(x)=xg(x)$.
Is it possible to conclude that $g$ is continuous everywhere except at zero?
It can also be easily shown that $g$ is continuous always imply the continuity of $f$.
Thank you for letting me know that $g(x)$ is not necessarily continuous at zero by counterexamples.
We assume that $x>0$ i.e. positive-semi-definite
On
No. There will always be a non-continuous $g$ with that property
Suppose that $f(x)=xg(x)$. If $g(x)$ is not continuous we're done. If $g(x)$ is continuous define $g'(x)=g(x)$ whenever $x\not = 0$ and $g'(0)=a$ for any $a$ such that $g(0)\not = a$. Then $g'$ is not continuous but $f(x)=xg'(x)$.
In other words you can always change the value of $g(x)$ at zero so you can't hope that $g$ will always be continuous.
Let $f(x):\equiv0$, and $g(x):=0$ $(x\ne0)$, $\>g(0):=1$.