A function f of \(\mathbb{R}\) in \(\mathbb{R}\) is a function of the greatest integer that is less than or equal to x if and only if : ∀x ∈ [n, n + 1] : x → [x] = n where n ∈ \(\mathbb{Z}\).
The greatest integer function is a synonym for a floor function.
Symbol
The function of the greatest integer that is less than or equal to x is noted as [x] and is read as “floor of x“.
We sometimes use the notation ⌊x⌋ to indicate the greatest integer less than or equal to, as opposed to the notation ⌈x⌉ used to refer to the least integer that is greater than or equal to.
Examples
- Here is a graph of the function of the greatest integer less than or equal to.
The small circle “ο” at the end point of each floor signifies that the limit point of the floor does not belong to the graph of this function.
- We want to find out the number of teams of 5 players that we can form with 17 candidates.
Because each team must have 5 players, we can only form 3 teams: f(17) = [17 ÷ 5] = 3.