Limits, Continuity & Differentiability
Discontinuity of Floor Function
nta_pyq_2024_apr
Grade 12
Question:
Let $f:[-1,2]\to\mathbb{R}$ be given by $f(x)=2x^2+x+[x^2]-[x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is:
Step-by-Step Solution
Key Concept: $f(x)=(2x^2+x-[x])+[x^2]$. The term $(2x^2+x-[x])$ is continuous (as $x-[x]=\{x\}$ can cause issues). $[x^2]$ is discontinuous at $x=\pm1,\sqrt{2},\sqrt{3},2$. Check each suspect point.
Checking all suspect points: discontinuous at $x=-1,0,\sqrt{2},\sqrt{3}$. Total 4 points.
Correct Answer: 4