The definition of an increasing function is a function f such that for all x and y, x < y implies f(x) <= f(y). For a strictly increasing function change <= to

The definition of an increasing function is a function f such that for all x and y, x < y implies f(x) <= f(y). For a strictly increasing function change <= to <. Your argument fails because you drop the antecedent in: x < f(x) => f(x) < f(f(x)). If f(0) = 0, you don’t have 0 < f(0) so can’t require that f(f(0))<f(0). Note however that it is easy to show that f must be strictly increasing if it is increasing for if x is not y and f(x) = f(y) then 3x = f(f(x)) = f(f(y)) = 3y which cannot be the case.