Question:
The function \( f(x) = [x]\cos\left( \frac{2x-1}{2}\pi \right) \), where \( [\,\cdot\,] \) denotes the greatest integer function, is discontinuous at
The function \( f(x) = [x]\cos\left( \frac{2x-1}{2}\pi \right) \), where \( [\,\cdot\,] \) denotes the greatest integer function, is discontinuous at
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If a discontinuous function is multiplied by a non-zero continuous function, discontinuity remains.
Updated On: Apr 23, 2026
- all \( x \)
- no \( x \)
- all integral points
- \( x \) which is not an integer
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The Correct Option is C
Solution and Explanation
Concept:
Greatest integer function is discontinuous at integer values.
Step 1: Analyze components. \[ [x] \text{ is discontinuous at integers} \] \[ \cos\left( \frac{2x-1}{2}\pi \right) \text{ is continuous everywhere} \]
Step 2: Product behavior: Discontinuity of \( [x] \) dominates unless multiplied by zero. At integers: \[ \cos\left(\frac{2n-1}{2}\pi\right) = \cos\left((n-\tfrac{1}{2})\pi\right) \neq 0 \] Final Answer: \[ \text{Discontinuous at all integers} \]
Step 1: Analyze components. \[ [x] \text{ is discontinuous at integers} \] \[ \cos\left( \frac{2x-1}{2}\pi \right) \text{ is continuous everywhere} \]
Step 2: Product behavior: Discontinuity of \( [x] \) dominates unless multiplied by zero. At integers: \[ \cos\left(\frac{2n-1}{2}\pi\right) = \cos\left((n-\tfrac{1}{2})\pi\right) \neq 0 \] Final Answer: \[ \text{Discontinuous at all integers} \]
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