Given any positive integers
$m$ and
$d$, we say a sequence of points
$(x_{i})_{i\in I}$ in
$\mathbb{R}^{m}$ is Lipschitz-
$d$-controlling if one can select suitable values
$y_{i}\;(i\in I)$ such that for every Lipschitz function
$f\,:\,\mathbb{R}^{m}\,\rightarrow \,\mathbb{R}^{d}$ there exists
$i$ with
$|f(x_{i})\,-\,y_{i}|\,<\,1$. We conjecture that for every
$m\leqslant d$, a sequence
$(x_{i})_{i\in I}\subset \mathbb{R}^{m}$ is
$d$-controlling if and only if
$$\begin{eqnarray}\displaystyle \sup _{n\in \mathbb{N}}\frac{|\{i\in I:|x_{i}|\leqslant n\}|}{n^{d}}=\infty . & & \displaystyle \nonumber\end{eqnarray}$$ We prove that this condition is necessary and a slightly stronger one is already sufficient for the sequence to be
$d$-controlling. We also prove the conjecture for
$m=1$.