# Private Algorithms Can be Always Extended

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

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