JEE Main 2026 Jan 22 Shift 2, Mathematics Q15: Continuity of Function

JEE Main2026Jan 22, Shift 2Mathematics

Let $[∙]$ denote the greatest integer function, and let $f (x) = min {2 x, x^{2}}$ . Let $S = {x \in (- 2, 2) :$ the function $g (x) = ∣ x ∣ [x^{2}]$ is discontinuous at $x}$ . Then $x \in S \sum f (x)$ equals :

A
$2 - 2$
B
$26 - 32$
C
$1 - 2$
D
$6 - 22$

Solution

$[x^{2}]$ jumps at integer values of $x^{2}$ within $(0, 4)$ , i.e., $x^{2} = 1, 2, 3$ , giving $x = \pm 1, \pm 2, \pm 3$ .

At $x = 0$ , $∣ x ∣ \cdot [x^{2}] = 0$ and the limit is $0$ , so $g$ is continuous there. Checking the six points above, $g$ is discontinuous at all of them. So $S = {- 1, 1, - 2, 2, - 3, 3}$ .

For $f (x) = min (2 x, x^{2})$ : when $x < 0$ , $2 x < 0 < x^{2}$ , so $f (x) = 2 x$ . When $x > 0$ , $f (x) = 2 x$ if $2 x \leq x^{2}$ (i.e., $x \geq 2$ ), else $x^{2}$ .

Evaluating: $f (- 1) = - 2$ , $f (1) = 1$ , $f (- 2) = - 2$ , $f (2) = 2$ , $f (- 3) = - 6$ , $f (3) = 6$ .

Sum $= - 2 + 1 - 2 + 2 - 6 + 6 = 1 - 2$ .

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JEE Main 2026 Jan 22 Shift 2, Mathematics Q15: Continuity of Function

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