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
29
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 1DifferentiationEASY
Let y=[x+1]y=[x+1], −4<x<−3-4<x<-3 where [.][.] is the greatest integer function. What is the derivative of yy with respect to xx at x=−3.5x=-3.5?

[Q71 · Apr · 2022]

Example 2Definite IntegrationEASY
What is ∫nn+1(x−[x]) dx\int_n^{n+1}(x-[x])\,dx, where [⋅][\cdot] is the greatest integer function and nn is natural number?

[Q92 · Sep · 2025]

Example 3Definite IntegrationMODERATE
What is ∫02[x2] dx\int_{0}^{\sqrt{2}}[x^2]\,dx equal to (where [.] is greatest integer function)?

[Q82 · Apr · 2018]

Example 4Limits & ContinuityHARD
The left-hand derivative of f(x)=[x]sin⁡(πx)f(x) = [x]\sin(\pi x) at x=kx = k, where kk is an integer and [x][x] is the greatest integer function, is

[Q79 · Sep · 2017]

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

29 questions from the bank — paginated, with cart and Word-export support.

Drill the 29 questions

Related principles

Often combined with this one — drill these next if you found the examples above tractable.