Floor, ceiling and the space between

Inspired by a question posed in Problem 27327 from Gazeta Matematică, we investigate the precise ranges of functions dependent on a real parameter and defined via the floor of a real number. Our findings reveal that determining these ranges necessitates non-trivial tools, including the Kronecker approximation theorem. Our results emphasize the deep distinctions among irrational parameters and contrast the behavior of sub-unitary and supra-unitary rational parameters. We also propose several conjectures related to irrational numbers and supra-unitary rational ones, supported by extensive computations using Wolfram Mathematica.