Limits, Continuity & Differentiability
Greatest Integer Function
Grade 12
Question:
<p>If \(f(x) = [\tan^2 x]\), where \([x]\) is the greatest integer function, then</p>
<p>A. \(\lim_{x \to 0} f(x)\) does not exist</p>
<p>B. \(f(x)\) is continuous at \(x = 0\)</p>
<p>C. \(f(x)\) is not differentiable at \(x = 0\)</p>
<p>D. \(f'(0) = 1\)</p>
Step-by-Step Solution
Key Concept: The greatest integer function creates a step function where f(x) is constant between integer values of tan²x. Discontinuities occur where tan²x passes through integer values, and differentiability fails at these jump discontinuities and where tan²x is undefined.
<p><strong>Step 1:</strong> Identify where tan²x is defined and continuous.</p><p>tan²x is defined and continuous for all x ≠ (2n+1)π/2, where n ∈ ℤ.</p><p><strong>Step 2:</strong> Analyze the floor function behavior.</p><p>Since f(x) = [tan²x], the function has jump discontinuities wherever tan²x = n (positive integer). Between consecutive integers, f(x) is constant (hence differentiable with f'(x) = 0).</p><p><strong>Step 3:</strong> Determine discontinuity points.</p><p>f(x) is discontinuous at:</p><p>• x = (2n+1)π/2 (where tan x is undefined)</p><p>• All x where tan²x ∈ ℤ⁺ (jump discontinuities from the floor function)</p><p><strong>Step 4:</strong> Determine differentiability.</p><p>f(x) is non-differentiable at:</p><p>• All discontinuity points (from above)</p><p>• All points where tan²x is an integer (jump discontinuities)</p><p>∴ f(x) is differentiable only on intervals where tan²x remains strictly between consecutive integers, with f'(x) = 0 on these intervals.</p>
Correct Answer: B