We consider the following fundamental question on $e$-differential privacy. Consider an arbitrary $e$-differentially private algorithm defined on a subset of the input space. Is it possible to extend it to an $e'$-differentially private algorithm on the whole input space for some $e'$ comparable with $e$? In this note we answer affirmatively this question for $e'=2e$. Our result applies to every input metric space and space of possible outputs. This result originally appeared in a recent paper by the authors [BCSZ18]. We present a self-contained version in this note, in the hopes that it will be broadly useful.