Question

The function \( f(x) = [x]\cos\left( \frac{2x-1}{2}\pi \right) \), 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:

If a discontinuous function is multiplied by a non-zero continuous function, discontinuity remains.

Answer 1

The Correct Option is C

To determine the points of discontinuity of the function \(f(x) = [x]\cos\left( \frac{2x-1}{2}\pi \right)\), where \([\cdot]\) denotes the greatest integer function, we need to explore the properties of each component in the function.

1. The greatest integer function \([x]\) is known to be discontinuous at all integers. It moves from an integer to the next at integral points, which are points of discontinuity.
2. The cosine function \(\cos\left( \frac{2x-1}{2}\pi \right)\) is continuous everywhere since cosine is a continuous function over the real numbers.
3. The discontinuity in a composite function like \(f(x) = [x]\cos\left( \frac{2x-1}{2}\pi \right)\) is primarily due to the greatest integer function. Therefore, the points of discontinuity of \(f(x)\) are precisely where \([x]\) is discontinuous, i.e., at all integer points.

Thus, the given function \(f(x)\) will be discontinuous at all integral points.

Conclusion: The correct answer is that the function \(f(x)\) is discontinuous at all integral points.