The Well-Rounded Linear Function

Abstract

The generic linear function $ax+b$ of a real variable, with $a, b, x \in {\bf R}$, is usually evaluated as a scale function (product) followed by a translation (sum). Our main result shows that when such a function is variously combined with rounding functions (floor and ceiling), exactly 67 inequivalent rounded generic linear functions result, of which 38 are integer-valued and 29 are not. Several related results are also established, with elucidation of the relevant equivalence class structures.