Fill in the blanks so that the resulting statement is correct. Let `f(x) = [x + 2] sin((pi)/([x + 1]))`, where `[*]` denotes greatest integral function. The domain of f is ……….and the points of discontinuity of f in the domain are

Answer

To solve the problem, we need to determine the domain of the function \( f(x) = [x + 2] \sin\left(\frac{\pi}{[x + 1]}\right) \), where \([\cdot]\) denotes the greatest integer function (GIF).

### Step 1: Identify the components of the function
The function consists of two parts:
1. The greatest integer function \([x + 2]\)
2. The sine function \(\sin\left(\frac{\pi}{[x + 1]}\right)\)

### Step 2: Determine the domain of the sine function
The sine function is defined for all real numbers, but we need to ensure that the argument \([x + 1]\) does not equal zero because we cannot divide by zero.

### Step 3: Find when \([x + 1] = 0\)
The greatest integer function \([x + 1]\) equals zero when:
\[ 0 \leq x + 1 < 1 \implies -1 \leq x < 0 \]
Thus, the function is undefined for \(x\) in the interval \([-1, 0)\).

### Step 4: Determine the overall domain of \(f(x)\)
Since the function is undefined in the interval \([-1, 0)\), the domain of \(f(x)\) is:
\[ (-\infty, -1) \cup [0, \infty) \]

### Step 5: Identify points of discontinuity
The function \(f(x)\) is continuous everywhere in its domain except at points where the greatest integer function changes value. This occurs at integer values of \(x + 2\) and \(x + 1\).

1. The points where \([x + 1]\) changes value are at integers, specifically at \(x = -1\) (which is excluded from the domain) and at \(x = 0\) (which is included in the domain).
2. The function is also discontinuous at \(x = -1\) since it is not included in the domain.

### Conclusion
Thus, the points of discontinuity in the domain are:
- \(x = 0\)

### Final Answer
The domain of \(f\) is \((- \infty, -1) \cup [0, \infty)\) and the points of discontinuity of \(f\) in the domain are \(x = 0\).

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