Limits, Continuity & Differentiability
Continuity and Differentiability of Piecewise Floor Function
nta_pyq_2023_apr
Grade 12
Question:
Let $f:(-2,2)\to\mathbb{R}$ be defined by $f(x)=\begin{cases}x[x], & -2<x<0\\ (x-1)[x], & 0\leq x<2\end{cases}$ where $[x]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2,2)$ at which $y=|f(x)|$ is not continuous and not differentiable, then $m+n$ is equal to ________.
Step-by-Step Solution
Key Concept: Expand $f(x)$ using floor values on each sub-interval, then check $|f(x)|$ for continuity breaks and non-differentiability (corners).
$m=1$ (discontinuity at $x=-1$), $n=3$ (corners at $x=-1,0,1$). $m+n=4$.
Correct Answer: 4