Is the relation on the set of real functions $fRg$ iff exists $k > 0$ such that for all $x \in \mathbb{R}$, $f(x) + k < g(x + k)$ transitive?
I have proved that it's not reflexive, not symmetric, not antisymmetric via counterexamples. All of these cases seemed to me tricky, and involved looking for some two functions with a particular relationship between their behaviour. For instance, I proved that $R$ is not antisymmetric considering $f(x) = 2x$ and $g(x) = 2x + 2$. However I stumble on looking for a counterexample for the transitivity. The relation seems to be formulated in such general terms though, that I can hardly imagine proving that it's transitive; intuition tells me that there must be some functions with particular local properties which refute transitivity.