Testing holomorphy on curves (Q1936808)

Let \(S \subset\mathbb C^n\). We say that \(f: S \rightarrow \mathbb C \) is holomorphic if \(f\) is a restriction on \(S\) of a function holomorphic in some open neighborhood of \(S\). The main result is as follows: Let \(D \subset\mathbb C^n\) be a domain. Then there exists a continuous foliation \(E\) of \(D\) into one (real) dimensional curves, such that any \(C^1\) function on \( D\) which is holomorphic on each of the curves of \(E\), is holomorphic on \(D.\) The needed foliation is constructed as a homeomorphic image of the natural continuous foliation of \(D\) by segments parallel to the axis \(\text{Re\,} z_1\).