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

To determine the points of discontinuity for the function \( f(x) = [x]\cos\left(\frac{2x - 1}{2}\right)\pi \), where \([\,\cdot\,]\) denotes the greatest integer function, we need to examine the behavior of both components: \( [x] \) and \( \cos\left(\frac{2x - 1}{2}\right)\pi \).

1. The greatest integer function \([x]\) is discontinuous at all integer points. This is because for any integer \( n \), \([x]\) shifts from \( n-1 \) to \( n \) abruptly as \( x \) changes from just below \( n \) to exactly \( n \), causing a jump discontinuity.
2. The function \(\cos\left(\frac{2x - 1}{2}\right)\pi\) can also induce discontinuities, particularly where the inner argument shifts over its periodic boundary, but it is inherently continuous over its real domain. Therefore, it does not contribute additional discontinuities that are not aligned with changes in \( x \).
3. Combining these, the dominant discontinuity factor is the greatest integer function \([x]\) that causes discontinuities at all integer values of \( x \).

Thus, the function \(f(x)\) is discontinuous at all integer points, confirming the correct answer is the option: all integral points.