Consider the following for the next two items that follow:
Let `f(x)=[x]`, where [.] is the greatest integer function and `g(x)=sinx` be two real valued function over R.
Which of the following statements is correct?

Options

A. A) Both f(x) and g(x) are continuous at x = 0
B. B) f(x) is continuous at x = 0, but g(x) is not continuous at x = 0
C. C) g(x) is continuous at x = 0, but f(x) is not continuous at x = 0
D. D) Both f(x) and g(x) are discontinuous at x = 0

Answer

To solve the problem, we need to analyze the continuity of the functions \( f(x) = [x] \) (the greatest integer function) and \( g(x) = \sin x \) at \( x = 0 \).

### Step-by-Step Solution:

1. **Understanding the Functions**:
- The function \( f(x) = [x] \) gives the greatest integer less than or equal to \( x \). For example:
- \( [1.2] = 1 \)
- \( [0] = 0 \)
- \( [-1.5] = -2 \)
- The function \( g(x) = \sin x \) is a standard trigonometric function.

2. **Checking Continuity of \( f(x) \) at \( x = 0 \)**:
- We need to find the left-hand limit and right-hand limit at \( x = 0 \).
- **Left-hand limit**:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} [x] = [0^-] = -1 \]
- **Right-hand limit**:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} [x] = [0^+] = 0 \]
- The value of the function at \( x = 0 \):
\[ f(0) = [0] = 0 \]
- Since the left-hand limit \(-1\) is not equal to the right-hand limit \(0\), \( f(x) \) is not continuous at \( x = 0 \).

3. **Checking Continuity of \( g(x) \) at \( x = 0 \)**:
- The sine function is continuous everywhere, including at \( x = 0 \).
- Therefore, we can directly state:
\[ g(0) = \sin(0) = 0 \]
- The limit from both sides as \( x \) approaches \( 0 \) is:
\[ \lim_{x \to 0} g(x) = \sin(0) = 0 \]
- Thus, \( g(x) \) is continuous at \( x = 0 \).

4. **Conclusion**:
- We have determined that:
- \( f(x) \) is not continuous at \( x = 0 \).
- \( g(x) \) is continuous at \( x = 0 \).
- Therefore, the correct statement is:
- \( g(x) \) is continuous at \( x = 0 \), but \( f(x) \) is not.

### Final Answer:
The correct option is: **C. \( g(x) \) is continuous when \( x = 0 \), but \( f(x) \) is not.**