Suppose $f(t,x(t)) \in \mathbb{R}^m$ and $x(t) \in \mathbb{R}^n$ $(n \geq m)$ are continuous in their arguments and bounded, and that $f(t,0) = 0$. Further, assume that the partial derivatives of $f$ are piecewise continuous and bounded (uniformly in $t$), and $\dot{x}(t)$ is bounded and piecewise continuous.
1) Can we say that there exists a parameterization such that $f(t,x(t)) = A(t) x(t)$ for all $t$ and $A(t)$ is continuous and bounded?
2) Can we argue that the $A(t)$ has bounded, piecewise continuous derivative?
3) What conditions are necessary for such a parameterization?
Regarding 1), $A(t) = f(t,x(t))x^\top(t)/\|x(t)\|^2$ comes to mind. I believe its derivative would be piecewise continuous as well, but I seem to be having trouble putting everything together.