Let Q be the set of rational numbers. A function f : Q → Q is called aquaesulian if
the following property holds: for every x, y ∈ Q,
f(x + f(y)) = f(x) + y or f(f(x) + y) = x + f(y).
Show that there exists an integer c such that for any aquaesulian function f there are at most c different rational numbers of the form f(r) + f( − r) for some rational number r, and find the smallest possible value of c.
