In this work we study level sets of pseudoperiodic mappings from R^n to R^n-1, i.e., mapping that can be represented as a sum of a linear and a periodic map. An example of a stable map from R^3 to R^2 is constructed for which all regular level lines contain more than one unbounded component. It is proven that the number of unbounded components of a regular level line can not increase when the map undergoes a small perturbation. In a confirmation of a conjecture of V.I. Arnold a class of maps is singled out for which all regular level sets contain exactly one unbounded component.