Limits, Continuity & Differentiability
Discontinuity and Non-Differentiability of Floor + Absolute Value
nta_pyq_2025_apr
Grade Class 11
Let $[x]$ denote the greatest integer function, and let $m$ and $n$ respectively be the numbers of points, where the function $f(x) = [x] + |x-2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to:
Step-by-Step Solution
Key Concept: Write $f$ as a piecewise function by splitting at integers $-2,-1,0,1,2$ and at $x=2$ (where $|x-2|$ changes). Identify discontinuities from $[x]$ jumps and non-differentiable points from corners.
Piecewise analysis shows $f$ is discontinuous at $x\in\{-1,0,1,2\}$ ($m=4$) and non-differentiable at $x\in\{-1,0,1,2\}$ ($n=4$, same points since each jump creates non-differentiability). $m+n=8$.
<div class="key-concept"><strong>Key Concept:</strong> Write $f$ as a piecewise function by splitting at integers $-2,-1,0,1,2$ and at $x=2$ (where $|x-2|$ changes). Identify discontinuities from $[x]$ jumps and non-differentiable points from corners.</div>
<div class="trap-box"><strong>Trap:</strong> $[x]$ is discontinuous at $-1,0,1,2$ (4 points in $(-2,3)$) giving $m=4$. Non-differentiable points include all discontinuity points plus any corners of $|x-2|$ that don't coincide — $x=2$ is already counted. Thus $n=4$, $m+n=8$.</div>
Correct Answer: 8