Hartogs-type Theorems in Real Algebraic Geometry, II

Let f:X→ℝ be a function defined on a connected nonsingular real algebraic set X in ℝ^n with dimX≥ 2. We prove that f is a regular function whenever the restriction f|_C is a regular function for every algebraic curve C in X that is an analytic submanifold homeomorphic to the unit circle and has at most one singular point. We also have a suitable version of this result for X not necessarily connected.