Is there a term for functions where $f(x) - x$ is periodic?

Question

Title says it all really. I'm working with a type of function $f: \mathbb{R} \rightarrow \mathbb{R}$ with the property that $f(x+k) = f(x)+k$ for integers $k$, or equivalently, $f(x) - x$ is periodic with period $1$. Is there a name in common usage for this type of function? Or if not, what would be an appropriate name? "Semi-periodic"? "Periodically increasing"?

Answer 1

This is a $\mathbb{Z}$-equivariant map, where it is understood that $\mathbb{Z}$ acts on $\mathbb{R}$ by translation.