Let [x] denote the greatest integer function. Then match List-I with…

The correct answer is Option (3) → (A) - (II), (B) - (I), (C) - (III), (D) - (IV)

(A) \(|x-1|+|x-2|\)

This matches (II) is continuous everywhere.

Each modulus function is continuous everywhere, and so is their sum. It has corners at x = 1 and x = 2:

    \(\text{slope } -2 \text{ for } x \lt 1,\quad 0 \text{ for } 1 \lt x \lt 2,\quad 2 \text{ for } x \gt 2\)

So it is not differentiable at x = 1 and x = 2, and it does not fit (I).

(B) \(x-|x|\)

This matches (I) is differentiable everywhere except at x = 0.

    \(x - |x| = 0 \text{ for } x \ge 0\)
    \(x - |x| = 2x \text{ for } x \lt 0\)

At x = 0 the left-hand derivative is 2 and the right-hand derivative is 0, so it is not differentiable there; everywhere else it is a straight line.

(C) \(x-[x]\)

This matches (III) is not differentiable at x = 1.

    \( \displaystyle \lim_{x \to 1^{-}}\left(x - [x]\right) = 1 - 0 = 1, \quad f(1) = 1 - 1 = 0 \)

It is discontinuous at every integer, so it is not differentiable at x = 1. It fits none of (I), (II) and (IV).

(D) \(x|x|\)

This matches (IV) is differentiable at x = 1.

    \(x|x| = x^2 \text{ for } x \ge 0\)
    \(x|x| = -x^2 \text{ for } x \lt 0\)

Its derivative is 2x for x > 0 and −2x for x < 0, and both one-sided derivatives at 0 are 0. So it is differentiable everywhere (derivative 2|x|), including x = 1 and x = 0; it does not fit (I).

So the matching is (A) - (II), (B) - (I), (C) - (III), (D) - (IV).

Tip: When some items fit more than one entry, match the items with only one possible match first.

Note: Some List-I items fit more than one List-II entry. (A) fits (II) and also (III); (B) fits (I) and also (II) and (IV); (D) fits (II) and (IV). The question can still be solved because (C) x − [x] has only one possible match, (III), and neither (A) nor (D) can match (I), since (A) is not differentiable at x = 1 and (D) is differentiable at x = 0; Option (3) is the only option that respects all of these.