Find the domain of continuity of f(x) = ` sin^(-1) x - [x]` , [] represents greatest integer function .

Explanation

To find the domain of continuity of the function \( f(x) = \sin^{-1}(x) - [x] \), where \([x]\) represents the greatest integer function, we will analyze the continuity of each component of the function.

### Step 1: Determine the domain of \( \sin^{-1}(x) \)

The function \( \sin^{-1}(x) \) is defined for \( x \) in the interval \([-1, 1]\). Therefore, the domain of \( \sin^{-1}(x) \) is:
\[ D_1 = [-1, 1] \]

### Step 2: Determine the domain of the greatest integer function \([x]\)

The greatest integer function \([x]\) is defined for all real numbers \( x \). Therefore, the domain of \([x]\) is:
\[ D_2 = \mathbb{R} \]

### Step 3: Find the common domain

The overall domain of the function \( f(x) \) will be the intersection of the domains of \( \sin^{-1}(x) \) and \([x]\):
\[ D = D_1 \cap D_2 = [-1, 1] \]

### Step 4: Identify points of discontinuity

The function \( f(x) \) can be expressed as the difference of two functions. The function \( \sin^{-1}(x) \) is continuous on its domain \([-1, 1]\). However, the greatest integer function \([x]\) is discontinuous at every integer point.

Within the interval \([-1, 1]\), the integer points are \( -1, 0, \) and \( 1 \). We need to check the continuity of \( f(x) \) at these points:

- At \( x = -1 \):
- \( f(-1) = \sin^{-1}(-1) - [-1] = -\frac{\pi}{2} - (-1) = -\frac{\pi}{2} + 1 \)

- At \( x = 0 \):
- \( f(0) = \sin^{-1}(0) - [0] = 0 - 0 = 0 \)

- At \( x = 1 \):
- \( f(1) = \sin^{-1}(1) - [1] = \frac{\pi}{2} - 1 \)

The function \( f(x) \) is discontinuous at \( x = -1, 0, \) and \( 1 \) because the greatest integer function \([x]\) has jump discontinuities at these points.

### Step 5: Define the domain of continuity

Thus, the function \( f(x) \) is continuous on the open intervals between these discontinuities:
\[ \text{Domain of continuity} = (-1, 0) \cup (0, 1) \]

### Final Answer

The domain of continuity of \( f(x) = \sin^{-1}(x) - [x] \) is:
\[ (-1, 0) \cup (0, 1) \]