Let `f(x)={{:(,1+x,"if "x lt 0),(,1+[x]+sin x,0 le x lt pi//2),(,3,x ge pi//2):}`
Statement-1: F is a continuous on R-[1]
Statement-2: The greatest integer function is discontinuous at every integer point.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to analyze the function \( f(x) \) defined in the piecewise manner and check the continuity at the points of interest. The function is defined as follows:

\[ f(x) = \begin{cases} 1 + x & \text{if } x < 0 \\ 1 + [x] + \sin x & \text{if } 0 \leq x < \frac{\pi}{2} \\ 3 & \text{if } x \geq \frac{\pi}{2} \end{cases} \]

### Step 1: Check Continuity at \( x = 0 \)

1. **Left-hand limit as \( x \to 0^- \)**:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (1 + x) = 1 \]

2. **Right-hand limit as \( x \to 0^+ \)**:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (1 + [x] + \sin x) = 1 + [0] + \sin(0) = 1 + 0 + 0 = 1 \]

3. **Value of the function at \( x = 0 \)**:
\[ f(0) = 1 + [0] + \sin(0) = 1 + 0 + 0 = 1 \]

Since the left-hand limit, right-hand limit, and the functional value at \( x = 0 \) are all equal to 1, the function is continuous at \( x = 0 \).

### Step 2: Check Continuity at \( x = 1 \)

1. **Left-hand limit as \( x \to 1^- \)**:
\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (1 + [x] + \sin x) = 1 + [1] + \sin(1) = 1 + 1 + \sin(1) = 2 + \sin(1) \]

2. **Right-hand limit as \( x \to 1^+ \)**:
\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (1 + [x] + \sin x) = 1 + [1] + \sin(1) = 1 + 1 + \sin(1) = 2 + \sin(1) \]

Both limits are equal, but we need to check the functional value at \( x = 1 \):

3. **Value of the function at \( x = 1 \)**:
\[ f(1) = 1 + [1] + \sin(1) = 2 + \sin(1) \]

Since the left-hand limit and right-hand limit are equal but not equal to the functional value \( f(1) \), the function is discontinuous at \( x = 1 \).

### Step 3: Check Continuity at \( x = \frac{\pi}{2} \)

1. **Left-hand limit as \( x \to \frac{\pi}{2}^- \)**:
\[ \lim_{x \to \frac{\pi}{2}^-} f(x) = \lim_{x \to \frac{\pi}{2}^-} (1 + [x] + \sin x) = 1 + [\frac{\pi}{2}] + \sin(\frac{\pi}{2}) = 1 + 1 + 1 = 3 \]

2. **Right-hand limit as \( x \to \frac{\pi}{2}^+ \)**:
\[ \lim_{x \to \frac{\pi}{2}^+} f(x) = 3 \]

3. **Value of the function at \( x = \frac{\pi}{2} \)**:
\[ f\left(\frac{\pi}{2}\right) = 3 \]

Since the left-hand limit, right-hand limit, and the functional value at \( x = \frac{\pi}{2} \) are all equal to 3, the function is continuous at \( x = \frac{\pi}{2} \).

### Conclusion

The function \( f(x) \) is continuous on \( \mathbb{R} \setminus \{1\} \) since it is discontinuous only at \( x = 1 \).

### Statements Evaluation

- **Statement 1**: \( f \) is continuous on \( \mathbb{R} - \{1\} \) (True).
- **Statement 2**: The greatest integer function is discontinuous at every integer point (True).

Since both statements are true, and Statement 2 correctly explains Statement 1, we conclude:

**Correct Option**: Statement 1 is true, Statement 2 is true, and Statement 2 is the correct explanation for Statement 1.