As the title says, I am working on a problem and I would like to know if there is a relation between a function $f$ being integrable and the function $f(\lfloor x \rfloor)$ being integrable. One can show that
$$\int|f(x) - f(\lfloor x \rfloor)|dx + \int |f(x)|dx \geq \int|f(\lfloor x \rfloor)|dx $$
so if the first integral on the left converges, then $f(\lfloor x \rfloor)$ must be integrable. It is true that the integrand is bounded a.e, but more than that I have not been able to show.
Is the statement even true, and if so any hints on how to show this?