Question

The function \(f(x) = [x]\cos\left[\frac{2x - 1}{2}\right]\pi\), where \([\,\cdot\,]\) denotes the greatest integer function, is discontinuous at

• A. all \(x\)
• B. no \(x\)
• C. all integral points
• D. \(x\) which is not an integer

Hint:

Greatest integer function \([x]\) has jump discontinuities at integers.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Greatest integer function \([x]\) is discontinuous at integers.
Step 2: Detailed Explanation:
\(f(x) = [x]\cos\left[\frac{2x - 1}{2}\right]\pi\)
The factor \([x]\) is discontinuous at all integer points. The cosine factor is continuous everywhere. Hence \(f(x)\) is discontinuous at all integral points.
Step 3: Final Answer:
All integral points.