Question

Let \( f : [-1, 1] \to \mathbb{R} \) be defined by \( f(x) = \dfrac{x^2 + [\sin\pi x]}{1 + |x|}, \text{ where } [y] \text{ denotes the greatest integer less than or equal to } y. \) Then 
 

• A. \( f \) is continuous at \( -\frac{1}{2}, 0, 1 \)
• B. \( f \) is discontinuous at \( -1, 0, \frac{1}{2} \)
• C. \( f \) is discontinuous at \( -1, -\frac{1}{2}, 0, \frac{1}{2} \)
• D. \( f \) is continuous everywhere except at 0

Hint:

When working with piecewise functions involving floor or ceiling functions, check for discontinuities where the argument crosses integer boundaries.

Answer 1

The Correct Option is B

Step 1: Analyze the function's components.
The function involves both integer and sine functions, which have specific points of discontinuity. We need to check whether the function is continuous or discontinuous at certain points in the domain \( [-1, 1] \). Since the function involves the greatest integer function and the absolute value, these create potential discontinuities.

Step 2: Analyze discontinuity.
- At \( x = 0 \), the behavior of \( f(x) \) can be disrupted by the greatest integer function. This creates a discontinuity at this point.
- Similarly, at \( x = -1 \) and \( x = \frac{1}{2} \), discontinuities occur due to the floor function in the definition of \( f(x) \).

Step 3: Conclusion.
The correct answer is (B) \( f \) is discontinuous at \( -1, 0, \frac{1}{2} \).