Level sets of differentiable functions of two variables with non-vanishing gradient (Q1614696)

This article deals with level sets of a function \(f: \mathbb{R}^2\to \mathbb{R}\), whose gradient exists everywhere and is nowhere zero. The author proves that, in this case, in a neighborhood of each of its points the level set \(\{x\in\mathbb{R}^2: f(x)= c\}\) is homeomorphic either to an open interval or to the union of finitely many open segments passing through a point; moreover, this second case holds only at the points of a discrete set. The proof is based on some results of plane continua; in particular, it is proved that the level sets (extended by \(\infty\)) consist of Jordan curves.