Let $[\bullet]$ denote the greatest integer function, and let $f(x)=\min \left\{\sqrt{2} x, x^{2}\right\}$. Let $S=\{x \in(-2,2):$ the function $\mathrm{g}(\mathrm{x})=|\mathrm{x}|\left[\mathrm{x}^{2}\right]$ is discontinuous at x$\}$. Then $\sum_{x \in S} f(x)$ equals :

JEE Main · 2026 · 22 Jan 2026 · Shift 2 · Continuity and Differentiability

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

\sum_{x \in S} f (x)

equals :

Answer: (C)

The value

[x^{2}]

changes exactly where

x^{2}

is a positive integer, so

g (x) = ∣ x ∣ [x^{2}]

is discontinuous where

x^{2} \in {1, 2, 3}

(- 2, 2)

, namely

x = \pm 1, \pm 2, \pm 3

(at these points

∣ x ∣ \neq = 0

, so the jump survives). With

f (x) = min {2 x, x^{2}}

f (- 3) = - 6, f (- 2) = - 2, f (- 1) = - 2, f (1) = 1, f (2) = 2, f (3) = 6 .

The terms

\pm 6

and

- 2, + 2

cancel, leaving

x \in S \sum f (x) = 1 - 2 .

JEE Main · 2026 · 22 Jan 2026 · Shift 2 · Single correct option