A note on Darboux functions (Q1803981)

It is proven: (1) If \(G\) is a family of nowhere constant, continuous functions such that the cardinal successor \((\text{card }G)^ +<2^ \omega\) then there is a Darboux function \(f\) such that \(f+g\) is not Darboux whenever \(g\in G\); and (2) One can assign a real \(c(g)\) to every continuous, nowhere linear function \(g\) such that the union of the graphs of the functions \(x\to g(x)+c(g)\) does not contain a straight segment.