Conditions for the derivative of a function to be a proper map

Question

Let $f:U \to \mathbb{R}^{m}$ be a continuously differentiable function, where $U \subset \mathbb{R}^{n}$. In this case, the function \begin{equation*} Df:U \to L(\mathbb{R}^{n},\mathbb{R}^{m}) \end{equation*} is continuous. Could one formulate conditions on $f$, which would suffice to assure that $Df$ is a proper map? Here by proper I mean the preimage of every compact set is compact.