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

We show that if the gradient of f :R2→R exists everywhere and is nowhere zero, then in a neighbourhood of each of its points the level set {x ∈ R2: f (x) = c} is homeomorphic either to an open interval or to the union of finitely many open segments passing through a point. The second case holds only at the points of a discrete set. We also investigate the global structure of the level sets.  2002 Elsevier Science (USA). All rights reserved.