<p>If <span class="math">f(x) = [x] + \frac{1}{3 + \frac{2}{x + 3}}</span> (where <span class="math">[\cdot]</span> denotes the greatest integer function), which of the following are correct?</p>
<p>(A) f(x) is discontinuous at x = 1, 10, 15</p>
<p>(B) f(x) is continuous at x = n/3, where n is any integer</p>
<p>(C) <span class="math">\int_0^{2/3} f(x) \, dx = \frac{1}{3}</span></p>
Step-by-Step Solution
Key Concept: We must analyze the continuity of f(x) = [x] + 1/(3 + 2/(x+3)) by first simplifying the nested fraction, then determining where the greatest integer function creates discontinuities. The function is discontinuous at integer values of x where [x] jumps.
<p><strong>Step 1: Simplify the nested fraction</strong></p><p>f(x) = [x] + 1/(3 + 2/(x+3))</p><p>= [x] + 1/((3(x+3) + 2)/(x+3))</p><p>= [x] + 1/((3x + 11)/(x+3))</p><p>= [x] + (x+3)/(3x + 11)</p></p><p><strong>Step 2: Analyze discontinuities from [x]</strong></p><p>The greatest integer function [x] is discontinuous at all integer values of x. At x = 1, 10, 15 (all integers), [x] has jump discontinuities where the left and right limits differ by 1. Since [x] is discontinuous at these points, f(x) is discontinuous at x = 1, 10, 15.</p></p><p><strong>Step 3: Verify the rational part (x+3)/(3x+11) is continuous</strong></p><p>The rational part is continuous wherever 3x + 11 ≠ 0, i.e., x ≠ -11/3. This doesn't affect x = 1, 10, 15.</p></p><p><strong>Step 4: Check option (B): Continuity at x = n/3</strong></p><p>At x = n/3 (where n is integer), the rational part is continuous, BUT the [x] term still causes discontinuity if n/3 is not matched by continuity in [x]. Only when n/3 is NOT near an integer step does [x] create discontinuity. This statement is not universally true.</p></p><p><strong>Step 5: Check option (C): Integration</strong></p><p>For x ∈ [0, 2/3], we have [x] = 0 throughout (since 0 ≤ x < 1).</p><p>∫₀^(2/3) f(x)dx = ∫₀^(2/3) [0 + (x+3)/(3x+11)]dx = ∫₀^(2/3) (x+3)/(3x+11)dx</p><p>This integral evaluates to approximately 0.189, NOT 1/3 ≈ 0.333.</p></p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A