Level Sets of Differentiable Functions of Two Variables with Non-vanishing Gradient

We show that if the gradient of $f:{\text{RR}}^2\to \text{RR}$ exists everywhere and is nowhere zero, then in a neighbourhood of each of its points the level set $\{x\in {\text{RR}}^2:f(x)=c\}$ 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.