If `f(x)=[x] sin ((pi)/([x+1]))`, where [.] denotes the greatest integer function, then the set of point of discontiuity of f in its domain is

Options

A. Z
B. `Z-{-1,0}`
C. `R-[-1,0)`
D. none of these

Explanation

To solve the problem, we need to analyze the function \( f(x) = [x] \sin\left(\frac{\pi}{[x] + 1}\right) \), where \([x]\) denotes the greatest integer function.

### Step 1: Identify the domain of \( f(x) \)

The function \( f(x) \) is defined as long as the denominator \([x] + 1\) is not equal to zero. This means we need to find when \([x] + 1 = 0\):
\[ [x] + 1 = 0 \implies [x] = -1 \]
The greatest integer function \([x] = -1\) when \( -1 \leq x < 0 \). Therefore, \( f(x) \) is undefined for \( x \) in the interval \([-1, 0)\).

### Step 2: Identify points of discontinuity

Next, we need to check the points of discontinuity. The function \( f(x) \) can be discontinuous at points where \([x]\) changes its value, which occurs at integer values of \( x \).

1. The points where \([x]\) changes are at integers: \( \ldots, -2, -1, 0, 1, 2, \ldots \).
2. Since \( f(x) \) is undefined for \( x \) in the interval \([-1, 0)\), we need to consider the endpoints:
- At \( x = -1 \), \( f(x) \) is undefined.
- At \( x = 0 \), \( f(x) \) is also undefined.

### Step 3: Conclusion

Thus, the points of discontinuity of \( f(x) \) are:
- The integers \( -1 \) and \( 0 \) where the function is not defined.

Therefore, the set of points of discontinuity of \( f \) in its domain is:
\[ \mathbb{Z} \setminus \{-1, 0\} \]

### Final Answer

The correct option is:
**Option 2: \( \mathbb{Z} \setminus \{-1, 0\} \)**

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