A well-rounded linear function (Q5942560)

There are 243 ways of combining the linear function \(ax+ b\) with the floor and ceiling function: each of the quantities \(a\), \(x\), \(ax\), \(b\), \(ax+ b\) can be rounded up, rounded down or not rounded. Two such functions are called equivalent if they have the same value for every ordered triple \((a,x,b)\) of real numbers. This article divides these 243 functions into 67 equivalence classes of which 38 are integer-valued. Several related results are also established, including a similar partition into 5 equivalence classes if \(a\) and \(b\) are required to be integers.