OIMU 2009 Problem 4

Given two positive integers $m,n$ , we say that a function $f : [0,m] \to \mathbb{R}$ is $(m,n)$ -slippery if it has the following properties:

i) $f$ is continuous; ii) $f(0) = 0$ , $f(m) = n$ ; iii) If $t_1, t_2\in [0,m]$ with $t_1 < t_2$ are such that $t_2-t_1\in \mathbb{Z}$ and $f(t_2)-f(t_1)\in\mathbb{Z}$ , then $t_2-t_1 \in \{0,m\}$ .

Find all the possible values for $m, n$ such that there is a function $f$ that is $(m,n)$ -slippery.