Algebraic surfaces determine analyticity of functions

Let $$f :X \rightarrow \mathbb {R}$$ f : X → R be a function defined on a nonsingular real algebraic set X of dimension at least 3. We prove that f is an analytic (resp. a Nash) function whenever the restriction $$f|_{S}$$ f | S is an analytic (resp. a Nash) function for every nonsingular algebraic surface $$S \subset X$$ S ⊂ X whose each connected component is homeomorphic to the unit 2-sphere. Furthermore, the surfaces S can be replaced by compact nonsingular algebraic curves in X, provided that dim$$X \ge 2$$ X ≥ 2 and f is of class $$\mathcal {C}^{\infty }$$ C ∞ .