Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>Graph of <span class="math-inline">\(f(x)\)</span> shown (piecewise linear on [0,5] with values f(0)=1, f(1)=1, f(2)=2, f(3)=2, f(4)=2, f(5)=2 approximately). Which are correct?</p>
non-removable disc at 2 pts
non-diff at 3 pts in domain
lim f(f(x))=1 as x\to 1
#disc = #non-diff
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>From the graph description: f has a jump discontinuity at x=1 (open circle at (1,2) and closed at (1,1)), linear segments otherwise.</p><p>(A) Non-removable discontinuity at two points: Only at x=1 (jump). Not two — FALSE.</p><p>(B) Non-differentiable at three points in domain: At x=1 (discontinuity), x=2 (corner), x=3 (corner) — TRUE ✓</p><p>(C) lim_{x→1} f(f(x))=1: f(x)→1 as x→1⁻, then f(1)=1. TRUE ✓</p><p>(D) Points of discontinuity = points of non-differentiability: 1≠3 — FALSE.</p><p><strong>Answer: (B),(C)</strong></p><div class="key-concept"><strong>Key Concept:</strong> Every discontinuity is a point of non-differentiability, but not vice versa</div></div>
Correct Answer: B,C