Some characterizations of the floor, ceiling and fractional part functions with related results

One of the most important issues for the frequent special functions is the uniqueness conditions of such functions. As far as we know, there are no characterizations for the floor, ceiling, and fractional part functions in general (as real functions $f:\mathbb{R}\to \mathbb{R}$). As integer-valued functions $f:\mathbb{Q}\to \mathbb{Z}$, we have found a just common characterization for the floor and ceiling functions. Motivated by the topic of decomposer functions and factors of groups, we introduce and prove some characterizations of the above-mentioned functions. We also state and prove the relevant results and some of their generalities and developments.