Continuity and Differentiability
GIF Discontinuity — Sum over Discontinuous Points
nta_pyq_2026_jan
Grade 12
Question:
Let $[\cdot]$ denote the greatest integer function, and let $f(x)=\min\{\sqrt{2}x,\,x^2\}$. Let $S=\{x\in(-2,2): g(x)=|x|\,[x^2]$ is discontinuous at $x\}$. Then $\displaystyle\sum_{x\in S}f(x)$ equals
$\sqrt{6}-2\sqrt{2}$
$1-\sqrt{2}$
$2-\sqrt{2}$
$2\sqrt{6}-3\sqrt{2}$
Step-by-Step Solution
Key Concept: $g(x)=|x|[x^2]$ is discontinuous where $[x^2]$ jumps, i.e., at $x=\pm1,\pm\sqrt{2},\pm\sqrt{3}$ in $(-2,2)$. So $S=\{-\sqrt{3},-\sqrt{2},-1,1,\sqrt{2},\sqrt{3}\}$.
$\sum_{x\in S}f(x)=1-\sqrt{2}$.
Correct Answer: 2