Behavior of continuous functions with respect to intersection patterns (Q1332580)

Starting from Čech's theorem asserting that for any real continuous function \(f\) on an interval \(I\), whose level sets are all finite, there exists a subinterval of \(I\), where \(f\) is monotonic, various relationships are described between the behavior (in terms of differentiability, monotonicity and bounded variation) of continuous functions and their intersections with lines and other families of functions. Old and recent results, belonging to S. Banach, S. Minakshisundaram, K. M. Garg, S. Agronsky -- A. M. Bruckner -- M. Laczkovich -- D. Preiss, D. Menshoff, N. Luzin, H. Whitney, A. M. Bruckner -- K. M. Gar, A. M. Bruckner -- J. Haussermann are discussed in this respect and the paper ends with the following theorem: The typical continuous function \(f\) is one-to-one almost everywhere; more precisely, there exists a set \(Z\) of Hausdorff dimension zero such that \(f\) is one- to-one on the complement of \(Z\). Relevant comments are concerned with J. Gillis' continuous functions (whose level sets are all perfect) and with A. S. Besicovitch's continuous functions (without unilateral derivatives, even infinite ones, at any point). Remarks of the reviewer. Related to Čech's theorem is the theorem of \textit{A. Marchaud} [Fundam. Math. 20, 105-116 (1933; Zbl 0007.03001)] asserting that any real continuous function whose level sets are finite is differentiable a.e. and becomes a function of bounded variation if its values are suitably modified on a set of arbitrarily small measure. The reviewer [C. R. Acad. Sci., Paris 244, 2345-2347 (1957; Zbl 0077.273)] has given a generalization of this theorem.