Limits, Continuity & Differentiability
Discontinuity of Floor-Plus-Fractional-Part Function
nta_pyq_2023_apr
Grade 12
Question:
Let $[x]$ be the greatest integer $\leq x$. Then the number of points in the interval $(-2,1)$ where the function $f(x)=|[x]|+\sqrt{x-[x]}$ is discontinuous, is _____.
Step-by-Step Solution
Key Concept: The fractional part $\sqrt{\{x\}}$ is continuous; $|[x]|$ jumps at integers. Discontinuities occur where both terms together create a jump.
Discontinuous at $x=-1$ and $x=0$. Count $=2$.
Correct Answer: 2