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Floor and Ceiling Rules

Proposition: For every x in R, we have:

x-1 < floor(x) <= x <= ceiling(x) < x+1,

floor(x) = x <=> x in Z <=> ceiling(x) = x,

floor(-x) = -ceiling(x), ceiling(-x) = -floor(x).

1. Floor lies on or below the first diagonal; ceiling lies above diagonal; if we shift diagonal down one unit, it lies completely below the floor; if we shift it up one unit, it lies completely above the ceiling.
2. Floor and ceiling intersect each other at the diagonal.
3. Floor and ceiling are reflections of each other about both axes.