If `f (x) = {{:([x], , 0 le x lt 3),(|[x]+1|, , - 3 lt x lt 0):},` where [.] denotes greatest integer function, then number of point at which `f (x)` is discontinuous in `(-3,3),` is equa to ________.

Explanation

To determine the number of points at which the function \( f(x) \) is discontinuous in the interval \((-3, 3)\), we will analyze the function piece by piece.

The function is defined as follows:
\[ f(x) = \begin{cases} 0 & \text{for } 0 \leq x < 3 \\ |[x] + 1| & \text{for } -3 < x < 0 \end{cases} \]
where \([x]\) denotes the greatest integer function.

### Step 1: Identify the intervals and their corresponding function values
1. **For \( -3 < x < 0 \)**:
- We will break this interval into smaller segments based on the values of \([x]\):
- **For \( -3 < x < -2 \)**: Here, \([x] = -3\)
\[ f(x) = |-3 + 1| = | -2 | = 2 \]
- **For \( -2 \leq x < -1 \)**: Here, \([x] = -2\)
\[ f(x) = |-2 + 1| = | -1 | = 1 \]
- **For \( -1 \leq x < 0 \)**: Here, \([x] = -1\)
\[ f(x) = |-1 + 1| = | 0 | = 0 \]

2. **For \( 0 \leq x < 3 \)**:
- Here, \( f(x) = 0 \).

### Step 2: Identify points of discontinuity
Now we will check the points where the function changes its definition or where the greatest integer function changes its value:
- The critical points in the interval \((-3, 3)\) are \( -3, -2, -1, 0, 1, 2 \).

### Step 3: Evaluate continuity at each critical point
1. **At \( x = -3 \)**:
- Left limit: Not defined (as \( x \) approaches \(-3\) from the left)
- Right limit: \( f(-3) = 2 \)
- **Discontinuous**.

2. **At \( x = -2 \)**:
- Left limit: \( f(x) = 2 \) (as \( x \) approaches \(-2\) from the left)
- Right limit: \( f(-2) = 1 \)
- **Discontinuous**.

3. **At \( x = -1 \)**:
- Left limit: \( f(-1) = 0 \)
- Right limit: \( f(-1) = 0 \)
- **Continuous**.

4. **At \( x = 0 \)**:
- Left limit: \( f(0) = 0 \)
- Right limit: \( f(0) = 0 \)
- **Continuous**.

5. **At \( x = 1 \)**:
- Left limit: \( f(1) = 0 \)
- Right limit: \( f(1) = 0 \)
- **Continuous**.

6. **At \( x = 2 \)**:
- Left limit: \( f(2) = 0 \)
- Right limit: Not defined (as \( x \) approaches \( 2 \) from the right)
- **Discontinuous**.

### Step 4: Count the points of discontinuity
From the analysis, we found that \( f(x) \) is discontinuous at:
- \( x = -3 \)
- \( x = -2 \)
- \( x = 2 \)

Thus, the total number of points at which \( f(x) \) is discontinuous in the interval \((-3, 3)\) is **3**.

### Final Answer
The number of points at which \( f(x) \) is discontinuous in \((-3, 3)\) is **3**.