Limits, Continuity & Differentiability
Discontinuity and non-differentiability of greatest integer functions
Grade 12
<p>The number of points at which <span>\([x^2]\)</span> and <span>\(a^{[x^2]}\)</span> are discontinuous and not differentiable in the interval <span>\(1 < x < 3\)</span> is:</p>
Step-by-Step Solution
Key Concept: A function is discontinuous/non-differentiable at points where the greatest integer function [·] jumps. For [x²], jumps occur when x² crosses integer values; for a^[x²], jumps occur at the same points plus we must check the base a. The total count requires identifying all integer values that x² attains in (1,3).
<p><strong>Step 1:</strong> Find the range of x² for x ∈ (1,3).</p><p>When x ∈ (1,3), we have x² ∈ (1,9).</p><p><strong>Step 2:</strong> Identify integer values in (1,9).</p><p>The integers are: 2, 3, 4, 5, 6, 7, 8.</p><p>That's 7 integer values.</p><p><strong>Step 3:</strong> Determine discontinuity points.</p><p>[x²] is discontinuous when x² equals an integer. For each integer n ∈ {2,3,4,5,6,7,8}, there exists x = √n in (1,3) where x² = n.</p><p>At each x = √n, [x²] jumps from (n-1) to n, creating a discontinuity.</p><p><strong>Step 4:</strong> Check a^[x²].</p><p>Since a^[x²] is a composition where [x²] is discontinuous at these 7 points, a^[x²] inherits these discontinuities (assuming a > 0, a ≠ 1).</p><p><strong>Step 5:</strong> Verify non-differentiability.</p><p>At points where [x²] is discontinuous, both functions fail to be continuous, hence not differentiable.</p><p>∴ Answer: <strong>7</strong></p>
Correct Answer: 7