Principle deep dive
Greatest integer / floor behaviour
[x] = n on [n, n+1), jumping by 1 at each integer — the discrete cousin of modulus. Every NDA trap lives at the integer boundary: one-sided limits, the derivative that is 0 on the interior and undefined at the ends, and integrals that split interval by interval. Spans Limits & Continuity, Definite Integration, Functions, Differentiation and Apps of Integration.
- questions in the bank
- 31
- tagged HARD
- 38%
- chapter spread
- 5
- worked examples below
- 4
When to reach for it
The expression contains [x], ⌊x⌋, or the words 'greatest integer' / 'integral part'.
Why this principle matters
[x] is the largest integer not exceeding x: [2.7] = 2, [3] = 3, and — the one students get wrong — [−2.7] = −3, not −2. It is constant on each interval [n, n+1) and jumps by exactly 1 at every integer. Think of it as the modulus's discrete cousin: also piecewise, but with jumps instead of a corner.
Everything NDA asks lives at the integer boundary. Approaching an integer from the left and the right gives values differing by 1, so the two-sided limit never exists there. The derivative is 0 on every open interval (n, n+1) — the function is flat — and undefined at the integers themselves. A derivative-of-floor question asked strictly between two integers is therefore almost always answered 0, and that is the whole question.
The fractional part {x} = x − [x] is the other half of the same identity, and it is periodic with period 1. That periodicity is what makes ∫ from n to n+1 of (x − [x]) dx the same for every n. For integrals of [x] or [x²] over a range, split the range at each integer where the floor value changes, evaluate a constant on each piece, and add — for [x²] the breakpoints are at √2, √3, 2 and so on, not at the integers.
4 worked examples from the bank
Each example demonstrates the principle on a real past-year question. Click to reveal the answer, then the solution.
Example 1 · Differentiation · Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
Example 2 · Definite Integration · Integration of Absolute Value, Piecewise, and Greatest Integer Functions
Example 3 · Definite Integration · Integration of Absolute Value, Piecewise, and Greatest Integer Functions
Example 4 · Limits & Continuity · One-Sided Limits, Greatest Integer, and Absolute Value Limits
Variants to recognise
Same principle, different surfaces. Pattern-match these on test day.
Negative arguments
[−2.7] = −3. The floor goes DOWN, away from zero, for negatives. The single most common slip.
One-sided limits at an integer
lim x→n⁻ [x] = n − 1 and lim x→n⁺ [x] = n. They differ by 1, so the two-sided limit never exists at an integer.
Derivative: 0 inside, undefined at the ends
Constant on (n, n+1) ⇒ derivative 0 there. At each integer there is a jump, so no derivative exists.
Fractional part {x} = x − [x]
Periodic with period 1 and always in [0, 1). Turns an awkward floor integral into the same integral on every unit interval.
Splitting ∫[f(x)] dx
Break the range wherever the floor value changes. For [x²] on [0, 2] the breakpoints are √2 and √3, not the integers.
Drill every greatest integer / floor behaviour question
31 questions from the bank — paginated, with cart and Word-export support.
Related principles
Often combined with this one — drill these next if you found the examples above tractable.