Greatest Integer Function Properties
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Direct answer
The greatest integer function [x] floors x to the largest integer not exceeding it: [x] = n precisely when n ≤ x < n + 1, so [2.7] = 2, [−2.1] = −3 (not −2), and [5] = 5. Its domain is all reals and its range is the integers; its graph is a staircase with jumps at every integer, continuous from the right with jump discontinuities of size 1 at each integer. The companion function is the fractional part {x} = x − [x], always in [0, 1), so every real number splits as x = [x] + {x}. The algebra worth memorising: [x + n] = [x] + n for integer n; [x] + [−x] = 0 if x is an integer and −1 otherwise; and the Hermite identity [x] + [x + 1/n] + ... + [x + (n−1)/n] = [nx].
What you must remember
- Definition discipline: [x] ≤ x < [x] + 1; for negatives, [−2.1] = −3 — flooring moves toward minus infinity, a source of most sign errors.
- Integer shift: [x + n] = [x] + n for any integer n; non-integer shifts have no clean formula and must be case-handled.
- The complement identity: [x] + [−x] = 0 if x ∈ Z, else −1; equally {x} + {−x} = 1 for non-integer x.
- Hermite's identity: [x] + [x + 1/n] + [x + 2/n] + ... + [x + (n−1)/n] = [nx]; the n = 2 case [x] + [x + 1/2] = [2x] is the one to know cold.
- Fractional part: {x} = x − [x] ∈ [0, 1); {x} = 0 exactly at integers; {x + n} = {x} for integer n (periodicity with period 1).
- Discontinuity structure: jump discontinuities at every integer (left limit [x] − 1... at integer n the left limit is n − 1, right limit and value both n), continuity elsewhere; [x] is non-decreasing everywhere.
- Integration and summation: ∫ from 0 to n of [x]dx = n(n − 1)/2, and Σ over k of [x + k/n]-type sums run through Hermite — the two places GIF meets calculus in JEE Main.
Solving a step-function equation
Solve [x]² − 5[x] + 6 = 0. The equation is genuinely algebraic in the integer [x]: factoring gives ([x] − 2)([x] − 3) = 0, so [x] = 2 or [x] = 3. Translating back through n ≤ x < n + 1: [x] = 2 means 2 ≤ x < 3, and [x] = 3 means 3 ≤ x < 4. The solution set is the interval [2, 4) — a single interval, not isolated points, because every x between 2 and just under 4 shares the same floor values. Notice the answer's shape: equations in [x] always resolve to unions of half-open intervals [n, n + 1), one for each integer root.
A graphing-flavoured companion: sketch y = [x/2]. As x runs over [0, 2), [x/2] = 0; over [2, 4), it equals 1; over [4, 6), 2 — steps of width 2 instead of 1, jumps at every even integer. Scaling inside the floor stretches the staircase horizontally by the reciprocal factor, and this stretch rule ([x/a] jumps at multiples of a) is the standard quick-sketch tool for composite GIF graphs in Advanced's graph-based items.
Endpoints and jumps
Two habits mark the fluent GIF user. First, negatives: [−0.3] = −1, and the identity [x] + [−x] = −1 for non-integers settles most sign disputes instantly — as in evaluating [x] + [−x] at x = 2.7: 2 + (−3) = −1. Second, endpoints: solutions live in half-open intervals, and writing x ∈ [2, 3] instead of [2, 3) includes x = 3, where [x] = 3 breaks the equation — the single most common lost mark in numerical-answer format. JEE Main tests computation: evaluate [x] at stated points, solve linear equations in [x], count discontinuities of [f(x)] for a given f (jumps occur where f crosses an integer). JEE Advanced tests composition and the calculus interface: limits of [x] at integers (left ≠ right, so no limit exists), ∫[x]dx over ranges, and equations mixing [x] with {x} such as [x]{x} = 6... type items solved by writing x = n + r with n ∈ Z and r ∈ [0, 1). The staircase never lies: sketch it before algebra whenever a composite appears.
Frequently asked questions
What is the definition of the greatest integer function?
[x] is the largest integer less than or equal to x, so [x] = n exactly when n ≤ x < n + 1; for example [2.7] = 2 and [−2.1] = −3.
What is the value of [x] + [−x]?
Zero when x is an integer, and −1 for every non-integer x — the identity that resolves most sign arguments.
What does Hermite's identity state?
[x] + [x + 1/n] + ... + [x + (n − 1)/n] = [nx]; the n = 2 case reads [x] + [x + 1/2] = [2x].
Where is the greatest integer function discontinuous?
At every integer, with a jump of 1: the left limit is n − 1 while the value and right limit are n; it is continuous everywhere else.
What is the solution set of [x]² − 5[x] + 6 = 0?
[x] = 2 or 3, giving x ∈ [2, 3) ∪ [3, 4), which merges into the single half-open interval [2, 4).