Limit, Continuity and Differentiability: JEE Main 2026 Mathematics question

Let $[\cdot]$ denote the greatest integer function, and let $f(x)=\min\{\sqrt2 x, x^2\}$. 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