Let [ · ] denote the greatest integer function, and let f ( x ) = min { 2 x , ? x 2 } . Let S = { x ( - 2 , 2 ) : the function g ( x ) = | x | [ x 2 ] is discontinuous at x } . Then x S f ( x ) equals: [2026]

Question

Let $[\cdot]$ denote the greatest integer function, and let $f (x) = \min {\sqrt{2 x}, x^{2}} .$

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: [2026]

Smart Achievers · Accepted Answer

(2)

$g (x) =$ $| x |$ $[x^{2}]$

$Points of discontinuity of g (x) in (- 2, 2) are (\pm 1, \pm \sqrt{2}, \pm \sqrt{3})$

$∴$ $S = {- 1, 1, - \sqrt{2}, \sqrt{2}, - \sqrt{3}, \sqrt{3}}$

$∵$ $f (x) = \min {\sqrt{2} x, x^{2}}$

$∴$ $\sum_{x \in S} f (x) = - \sqrt{2} + 1 - 2 + 2 - \sqrt{6} + \sqrt{6}$

$= 1 - \sqrt{2}$

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