### Abstract

There are two topics at the base of this talk: arc-analytic functions (mostly studied in the subanalytic setting) and the relationships between o-minimal structures and Hardy fields. They lead us to study the following problem: for a function definable in some o-minimal structure (over the field of reals), what kind of property may be detected in restriction to some "small" space of definable arcs. We shall prove for instance that for most classical polynomially bounded o-minimal structures, continuity is equivalent to continuity on restriction to polynomial arcs. We will finish with problems of arc-definability for which we will give several open questions.