FloorFunktion

FloorFunktion, commonly denoted ⌊x⌋ and also called the floor function, maps a real number x to the greatest integer less than or equal to x. In formal terms, ⌊x⌋ ∈ Z and ⌊x⌋ ≤ x < ⌊x⌋ + 1. Equivalently, ⌊x⌋ = max{ n ∈ Z | n ≤ x }. Examples: ⌊3.7⌋ = 3, ⌊-2.4⌋ = -3.

Its domain is the real numbers and its range is the integers. The floor function is nondecreasing and piecewise constant, producing a step-like graph with jumps at each integer. Key identities include ⌊x + k⌋ = ⌊x⌋ + k for any integer k, and ⌊x⌋ = x when x is an integer. The fractional part is defined by {x} = x − ⌊x⌋, satisfying 0 ≤ {x} < 1.

Relation to other functions: ⌈x⌉ = -⌊-x⌋ connects the floor and ceiling functions. For positive values, floor and truncation toward zero coincide, but for negative values they can differ; in programming languages, floor(x) and truncation toward zero often diverge.

Graphically, the FloorFunktion is a right-continuous step function with horizontal segments on intervals [n, n+1) for integers n, and jumps at each integer. It is widely used in mathematics and computer science for rounding down, counting, modular arithmetic, and algorithms that require integer results. It is also known as the greatest integer function in some texts.