Holomorphic functions on subsets of C

Abstract

Let Gamma be a C-infinity curve in C containing 0; it becomes Gamma(theta) after rotation by angle theta about 0. Suppose a C-infinity function f can be extended holomorphically to a neighborhood of each element of the family {Gamma(theta)}. We prove that under some conditions on Gamma the function f is necessarily holomorphic in a neighborhood of the origin. In case Gamma 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.