Hartogs-Type theorems in real algebraic geometry, I

Let f W X ! R be a function defined on a connected nonsingular real algebraic set X in Rn. We prove that regularity of f can be detected by controlling the restrictions of f to either algebraic curves or algebraic surfaces in X. If dimX 2 and k is a positive integer, then f is a regular function whenever the restriction f jC is a regular function for every algebraic curve C in X that is a Ck submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity. Moreover, in the latter case, the singularity of C is equivalent to the plane curve singularity defined by the equation xp D yq for some primes p q. If dimX 3, then f is a regular function whenever the restriction f jS is a regular function for every nonsingular algebraic surface S in X that is homeomorphic to the unit 2-sphere. We also have suitable versions of these results for X not necessarily connected.