Holomorphic functions on subsets of C

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