# Does a function with these properties exist?

For $$x_1, x_2, x_3 \in \mathbb{Z}^+$$, does there exist a function $$f(\cdot)$$ defined on $$\mathbb{Z}^+$$, not necessarily continuous or differentiable, such that:

$$f(x_1) > f(x_2) \\ f(x_2) > f(x_3) \\ f(x_3) > f(x_1)$$

My immediate thought is that no such function exists, since a function can only be a many-to-one or one-to-one relation, and the above would require a one-to-many relation. If so, is there a more formal way of showing that the above is impossible? Or is it trivial to observe?

If I am wrong and such a function does exist, what is an example of one?

If such a function exists, by transitivity, we would have $$f(x_1) > f(x_1)$$ which is a contradiction.