Limits, Continuity & Differentiability
Continuity and Differentiability
Grade 12
Question:
<p>If \(f(x) = \tan^2\left[x - \frac{\pi}{4}\right]\) where \([x]\) is the greatest integer function then:</p>
<p>(a) \(f(x)\) is continuous at \(x = 0\)</p>
<p>(b) \(f(x)\) is differentiable at \(x = \frac{\pi}{4}\)</p>
<p>(c) \(f(x)\) is continuous in \(\left(0, \frac{\pi}{2}\right)\)</p>
<p>(d) \(f(x)\) is differentiable in \(\left(0, \frac{\pi}{2}\right)\)</p>
Step-by-Step Solution
Key Concept: The greatest integer function [x] creates a piecewise constant behavior, so f(x) is constant on each interval [n, n+1). The key is evaluating tan²(n - π/4) for integer values, recognizing that tan(n - π/4) depends on the periodicity of tangent (period π).
<p><strong>Step 1:</strong> Recognize that [x] is the greatest integer function, so for x ∈ [n, n+1) where n is an integer, f(x) = tan²(n - π/4) is constant.</p><p><strong>Step 2:</strong> Since tan has period π, we have tan(n - π/4) takes specific values based on n mod π. For integer n: tan(n - π/4) cycles through values as n increases.</p><p><strong>Step 3:</strong> The function f(x) is discontinuous at every integer value of x (jump discontinuities) because [x] jumps. At non-integer points within each interval [n, n+1), f(x) is continuous (it's constant).</p><p><strong>Step 4:</strong> Since f is piecewise constant with jumps at integers, f is NOT differentiable at integer points. Between integers, f'(x) = 0 (the derivative of a constant).</p><p>∴ <strong>Answer: A</strong> (f is continuous on (n, n+1) but discontinuous at integers; f is differentiable only on intervals strictly between integers)</p>
Correct Answer: A