2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/216813For 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{é} inequality, we obtain that $D^{\infty}(X)=N^{1, \infty}(X)$.28 pages, 2 figuresMetric Geometry46E15; 46E35text