Limits, Continuity & Differentiability
Greatest integer function and continuity
Grade 12
Question:
<p>Let <span>\([x]\)</span> be the greatest integer function. The function <span>\(f(x) = \frac{\sin([x])}{[x]}\)</span> on the interval <span>\([0, \pi]\)</span> is:</p>
<p>(a) not continuous at any point</p>
<p>(b) continuous at <span>\(x = \frac{3}{2}\)</span></p>
<p>(c) discontinuous at <span>\(x = 2\)</span></p>
<p>(d) differentiable at <span>\(x = \frac{4}{3}\)</span></p>
Step-by-Step Solution
Key Concept: The greatest integer function is inherently discontinuous at every integer; functions defined using <span>$[x]$</span> inherit these discontinuities.
<p><strong>Analysis:</strong> The greatest integer function is discontinuous at every integer point. For <span>$x \in [0, \pi]$</span>, the function <span>$[x]$</span> takes values 0, 1, 2, 3. At integer points, <span>$[x]$</span> jumps, causing <span>$f(x) = \frac{\sin([x])}{[x]}$</span> to be discontinuous (undefined at <span>$x=0$</span> due to division by zero, and jumps at <span>$x = 1, 2, 3$</span>). At <span>$x = 2$</span>, there is a jump discontinuity.</p><p>∴ Answer is (c).</p>
Correct Answer: C