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JEE Mains Maths · Differentiation

Counting Points of Non-Differentiability

Counting the points where a function with a modulus, a maximum, a minimum or the greatest integer function fails to be continuous or differentiable.

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Why this matters

Twenty-one PYQs, eleven of them multiple choice, and three from 2026. Twelve look for corners of a modulus, a maximum or a minimum and check which candidates survive; nine involve the greatest integer function, where every jump is also a point of non-differentiability. Two ideas cover the page.

Concept 1 of 2: Corners of modulus, max and min

∣g(x)∣|g(x)| can have a corner only where g(x)=0g(x)=0; max⁡{g,h}\max\{g,h\} and min⁡{g,h}\min\{g,h\} only where g=hg=h. Each such point is a candidate, and it is a real corner only if the slopes differ there. A factor that vanishes at the same point removes the corner: (x−a)∣x−a∣(x-a)|x-a| is smooth. When several terms break at one point, add their slope jumps; they can cancel.

Definition

  • ∣g∣|g|: check where g=0g=0; a simple root is a corner.
  • ∣g∣|g| at a double root, like ∣(x−1)2∣|(x-1)^2|, is smooth.
  • max⁡{g,h}\max\{g,h\}, min⁡{g,h}\min\{g,h\}: check where g=hg=h and the slopes differ.
  • h(x) ∣x−a∣h(x)\,|x-a| is differentiable at aa when h(a)=0h(a)=0.
  • Several terms at one point: their slope jumps add, and may cancel.

Slope jump at a corner

k ∣x−a∣:f′(a+)−f′(a−)=2kk\,|x-a|:\quad f'(a^+)-f'(a^-)=2k

Worked example

At how many points is f(x)=∣x2−4∣−4∣x−2∣f(x)=|x^2-4|-4|x-2| not differentiable?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 4 Apr 2026 Shift 1 · Q75Moderate

Example 1 · Differentiation · Counting Points of Non-Differentiability

The number of points, at which the function f(x)=max{6x,2+3x2}+∣x−1∣cos⁡∣x2−14∣,x∈(−π,π)f(x) = max\left\{ 6x,2 + 3x^{2} \right\} + |x - 1|\cos\left| x^{2} - \frac{1}{4} \right|,x \in ( - \pi,\pi), is not differentiable, is ____\_\_\_\_ .

A candidate is not yet a corner

Every zero inside a modulus is only a candidate. ∣x−1∣sin⁡∣x−1∣|x-1|\sin|x-1| and e∣(x−1)2∣e^{|(x-1)^2|} are smooth at 1, and a factor that vanishes at the same point removes the corner. Test each candidate before counting it.
Drill 11 more on corners of modulus, max and min

Concept 2 of 2: Jumps of the greatest integer function

[g(x)][g(x)] is constant while gg stays between two integers, and it jumps where gg crosses an integer. A jump is a discontinuity, and a function that is not continuous is not differentiable, so every jump counts for both. A peak exactly at an integer also counts: 5sin⁡x5\sin x reaches 5 at π2\frac\pi2, so [5sin⁡x][5\sin x] is 5 there and 4 on both sides. A trough at an integer does not: [x2][x^2] is 0 on both sides of 0.

Definition

  • [g(x)][g(x)] jumps where gg crosses an integer, or peaks exactly at one.
  • Not continuous means not differentiable: each jump counts for both.
  • [x+n]=[x]+n[x+n]=[x]+n for an integer nn.
  • Write the function interval by interval between the integers, then test each break.
  • A continuous term cannot cancel a jump; only another jump at the same point can.

Greatest integer function

[x]=n  for  n≤x<n+1,[x+n]=[x]+n  (n∈Z)[x]=n\ \text{ for }\ n\le x<n+1,\qquad[x+n]=[x]+n\ \ (n\in\mathbb{Z})

Worked example

At how many points of (0,3)(0,3) is f(x)=[x2]f(x)=[x^2] not differentiable?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 28 Jan 2025 · Q74Moderate

Example 2 · Differentiation · Counting Points of Non-Differentiability

Let
f(x)={3x,x<0min⁡{1+x+[x],x+2[x]},0≤x≤25,x>2f(x) =\left\{ \begin{matrix} 3x, & x < 0 \\ \min\{ 1 + x + \lbrack x\rbrack,x + 2\lbrack x\rbrack\}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{matrix} \right.
where [.] denotes greatest integer function. If α\alpha and β\beta are the number of points, where f is not continuous and is not differentiable, respectively, then α+β\alpha+\beta equals. ………\ldots\ldots\ldots

A jump counts twice in m + n

When a question asks for the points where ff is not continuous (mm) and not differentiable (nn), a jump belongs to both counts. Add every jump to nn as well as to mm.
Drill 8 more on jumps of the greatest integer function

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Corners of modulus, max and min

    Slope jump at a corner

    k ∣x−a∣:f′(a+)−f′(a−)=2kk\,|x-a|:\quad f'(a^+)-f'(a^-)=2k
  • Jumps of the greatest integer function

    Greatest integer function

    [x]=n  for  n≤x<n+1,[x+n]=[x]+n  (n∈Z)[x]=n\ \text{ for }\ n\le x<n+1,\qquad[x+n]=[x]+n\ \ (n\in\mathbb{Z})

Watch out for (2)

Test yourself on Differentiation

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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