On Tribonacci functions and Tribonacci numbers

In this paper, we consider Tribonacci functions on the real numbers R; i.e, functions f : R -> R such that for all x is an element of R, f(x + 3) = f(x + 2) + f(x + 1) + f(x). We develop the notion of Tribonacci functions using the concept of f-even and f-odd functions. Moreover, we show that if f is a Tribonachi function, then lim(x ->infinity) f(x + 1)/f(x) = beta such that beta is one of the roots of equation x(3) - x(2) - x - 1 = 0 for which beta is greater than one.