Analytic continuation on complex lines

The following extension theorem is proved. Let $\Omega \subset {\mathbf {C}}$ be an open set containing $\Delta$, the open unit disc in ${\mathbf {C}}$, and the point 1. Suppose that $f$ is holomorphic on $B$, the open unit ball of ${{\mathbf {C}}^N}$, let $x \in \partial B$ and assume that for all $y \in \partial B$ in a neighborhood of $x$ the function $\varsigma \to f(\varsigma y)$, holomorphic on $\Delta$, continues analytically into $\Omega$. Then $f$ continues analytically into a neighborhood of $x$.