Limits, Continuity & Differentiability
Continuity of Composite Functions
Grade 12
Question:
<p>Find the number of points where <m>f(x) = [\sin x - \cos x]</m> (where <m>[\cdot]</m> denotes greatest integral function), <m>x \in [0, 2\pi]</m> is not continuous.</p>
Step-by-Step Solution
Key Concept: The greatest integer function is discontinuous at integer values. Find where the argument of the greatest integer function takes integer values.
<p><strong>Step 1:</strong> We know <m>[\cdot]</m> is not continuous at integral points.</p><p><strong>Step 2:</strong> Thus, <m>f(x) = [\sin x - \cos x]</m> will be discontinuous at those points where <m>\sin x - \cos x</m> is an integer.</p><p><strong>Step 3:</strong> The points where <m>\sin x - \cos x</m> is an integer in <m>[0, 2\pi]</m> are: <m>x = \frac{\pi}{4}, \frac{3\pi}{4}, \pi, \frac{3\pi}{2}, \frac{7\pi}{4}</m></p><p>∴ The number of points at which <m>f(x)</m> is discontinuous is <strong>5</strong>.</p>
Correct Answer: 5