Holomorphic functions on subsets of ℂ

Abstract

Let Γ be a C^∞ curve in ℂ containing 0; it becomes Γ_θ after rotation by angle θ about 0. Suppose a C^∞ function f can be extended holomorphically to a neighborhood of each element of the family {Γ_θ}. We prove that under some conditions on Γ the function f is necessarily holomorphic in a neighborhood of the origin. In case Γ is a straight segment the well known Bochnak-Siciak Theorem gives such a proof for real analyticity. We also provide several other results related to testing holomorphy property on a family of certain subsets of a domain in ℂ.