THE 68th ROMANIAN MATHEMATICAL OLYMPIAD

Question

a) Show that there exist functions f : R R and g : R R such that f g = g f, f f = g g and f(x) g(x), for all x R. b) Prove that if f : R R and g : R R are continuous and have the properties f g = g f and f(x) g(x), for any x R, then (f f)(x) (g g)(x), for all x R.

Accepted Answer

a) The functions defined by f(x) = cases -1, & x < 0 1, & x 0 cases, g(x) = cases 1, & x < 0 -1, & x 0 cases fulfill the given conditions. b) The function h = f - g is continuous and doesn't vanish, so h(x) > 0, for all x R, or h(x) < 0, for all x R. The case f(x) > g(x), for all x R, gives f(f(x)) > g(f(x)) = f(g(x)) > g(g(x)), for all x R. In the second case the proof is similar.