Continuous Rigid Functions

A function f : R→ R is vertically [horizontally] rigid for C ⊆ (0,∞) if graph(cf) [graph(f(c ·))] is isometric with graph(f) for every c ∈ C. f is vertically [horizontally] rigid if this applies to C = (0,∞). Balka and Elekes have shown that a continuous function f vertically rigid for an uncountable set C must be of one of the forms f(x) = px+q or f(x) = pe + r, this way confirming Jancović’s conjecture saying that a continuous f is vertically rigid if and only if it is of one of these forms. We prove that their theorem actually applies to every C ⊆ (0,∞) generating a dense subgroup of ((0,∞), ·), but not to any smaller C. A continuous f is shown to be horizontally rigid if and only if it is of the form f(x) = px + q. In fact, f is already of that kind if it is horizontally rigid for some C with card(C ∩ ((0,∞) \ {1})) ≥ 2.