  We consider the following fundamental question on $\epsilon$-differential
privacy. Consider an arbitrary $\epsilon$-differentially private algorithm
defined on a subset of the input space. Is it possible to extend it to an
$\epsilon'$-differentially private algorithm on the whole input space for some
$\epsilon'$ comparable with $\epsilon$? In this note we answer affirmatively
this question for $\epsilon'=2\epsilon$. 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.
