Limits, Continuity & Differentiability
Continuity of Functions
Grade 12
Question:
<p>The function \( f(x) = \left[ x^2 \left[ \dfrac{1}{x^2} \right] \right] \), \( x \neq 0 \) is ( [x] represents the greatest integer \( \leq x \))</p>
<p>(a) continuous at \( x = 1 \)</p>
<p>(b) discontinuous at \( x = -1 \)</p>
<p>(c) discontinuous at infinitely many points</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Analyze the behavior of [1/x²] for different ranges of x, then multiply by x² and apply the outer floor function. The critical insight is that [1/x²] takes discrete integer values depending on the magnitude of x.
<p><strong>Step 1:</strong> For x ≠ 0, determine the range of 1/x². As |x| increases from 0, 1/x² decreases from ∞.</p><p><strong>Step 2:</strong> Find [1/x²] for different intervals:</p><ul><li>When |x| ≤ 1: 1/x² ≥ 1, so [1/x²] ≥ 1</li><li>When 1 < |x| ≤ √2: 1/2 ≤ 1/x² < 1, so [1/x²] = 0</li><li>When √2 < |x| ≤ √3: 1/3 ≤ 1/x² < 1/2, so [1/x²] = 0</li><li>When |x| > 1: 1/x² < 1, so [1/x²] = 0</li></ul><p><strong>Step 3:</strong> Calculate x²[1/x²] for x ≠ 0:</p><ul><li>When |x| ≤ 1: x² · [1/x²] ranges from 0 to x² (depending on exact value of [1/x²])</li><li>When |x| > 1: x² · 0 = 0</li></ul><p><strong>Step 4:</strong> Apply outer floor function [x²[1/x²]]:</p><ul><li>For |x| > 1: f(x) = [0] = 0</li><li>For |x| = 1: f(x) = [1·1] = 1</li><li>For 0 < |x| < 1: f(x) takes various values based on precise interval</li></ul><p><strong>Conclusion:</strong> f(x) is discontinuous at x = 0 and exhibits jump discontinuities throughout its domain. It is neither continuous nor differentiable at any point.</p><p>∴ Answer: D</p>
Correct Answer: D