Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D ∞ ( X ) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D ∞ ( X ) is compared with the space LIP ∞ ( 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 ∞ ( X ) with the Newtonian–Sobolev space N 1 , ∞ ( X ) . In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D ∞ ( X ) = N 1 , ∞ ( X ) .