VERTICALLY RIGID FUNCTIONS.

A function f : ℝ → ℝ is said to be vertically rigid provided its graph G(f) = {〈x, f(x)〉: x ∈ R} is isometric with the graph of the function kf for every non-zero k ∈ ℝ. We show that the group homomorphisms f from 〈ℝ,+〉 into 〈ℝ+, ·〉 is vertically rigid if and only if it is an epimorphism. Some other examples of vertically rigid functions will also be given. A problem of characterizing all vertically rigid functions remains open. [ABSTRACT FROM AUTHOR]