  For a metric space $X$, we study the space $D^{\infty}(X)$ of bounded
functions on $X$ whose infinitesimal Lipschitz constant is uniformly bounded.
$D^{\infty}(X)$ is compared with the space $\LIP^{\infty}(X)$ of bounded
Lipschitz functions on $X$, in terms of different properties regarding the
geometry of $X$. We also obtain a Banach-Stone theorem in this context. In the
case of a metric measure space, we also compare $D^{\infty}(X)$ with the
Newtonian-Sobolev space $N^{1, \infty}(X)$. In particular, if $X$ supports a
doubling measure and satisfies a local Poincar{\'e} inequality, we obtain that
$D^{\infty}(X)=N^{1, \infty}(X)$.
