Number of points of non-differerentiable of `f(x)=sin pi(x-[x])"in "(-pi//2,[pi//2).` Where [.] denotes the greatest integer function is

Options

A. f(x) is discontinuous at `x = {-1, 0, 1}`
B. f(x) is differentiable for `x in (-(pi)/(2),(pi)/(2))-{0}`
C. f(x) is differentiable for `x in (-(pi)/(2),(pi)/(2)) - {-1, 0, 1}`
D. None of these

Answer

To find the number of points of non-differentiability of the function \( f(x) = \sin(\pi x - [x]) \) in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), where \([x]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Understand the function
The function involves the greatest integer function \([x]\), which is non-differentiable at integer points. The interval we are considering is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

### Step 2: Identify the integer points in the interval
The integer points within the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) are \( -1, 0, \) and \( 1 \). These points are where the greatest integer function \([x]\) changes its value.

### Step 3: Analyze the function around the integer points
1. For \( x \) in the interval \( (-1, 0) \):
- Here, \([x] = -1\), so \( f(x) = \sin(\pi x + 1) \).

2. For \( x \) in the interval \( (0, 1) \):
- Here, \([x] = 0\), so \( f(x) = \sin(\pi x) \).

3. For \( x \) in the interval \( (1, \frac{\pi}{2}) \):
- Here, \([x] = 1\), so \( f(x) = \sin(\pi x - 1) \).

### Step 4: Determine differentiability
The function \( f(x) \) will be non-differentiable at the points where \([x]\) changes its value, which are the integer points \( -1, 0, \) and \( 1 \).

### Step 5: Conclusion
Thus, the points of non-differentiability of \( f(x) \) in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) are \( -1, 0, \) and \( 1 \). Therefore, the number of points of non-differentiability is:

\[ \text{Number of points of non-differentiability} = 3 \]

### Final Answer
The number of points of non-differentiability of \( f(x) \) in the given interval is \( 3 \).

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