A non-DC function which is DC along all convex curves

A problem asked by the authors in 1989 concerns the natural question, whether one can deduce that a continuous function f on an open convex set D⊂Rn is DC (i.e., is a difference of two convex functions) from the behavior of f “along some special curves φ”. I.M. Prudnikov published in 2014 a theorem (working with convex curves φ in the plane), which would give a positive answer in R2 to our problem. However, in the present note we construct an example showing that this theorem is not correct, and thus our problem remains open in each Rn, n>1.