Consider the following statements:
I. The function `f(x)=[x]`. Where [.] is the greatest integer function defined on R. is continuous at all points except at x = 0.
II. The function `f(x)=sin|x|` is continuous for all `x inR`.
Which of the above statements is/are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To determine which of the statements is correct, we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement I:** The function \( f(x) = [x] \), where \([.]\) is the greatest integer function, is continuous at all points except at \( x = 0 \).

1. The greatest integer function \([x]\) returns the largest integer less than or equal to \( x \).
2. This function is discontinuous at every integer point. For example:
- At \( x = 1 \):
- Left-hand limit as \( x \) approaches 1 from the left: \( \lim_{x \to 1^-} [x] = 0 \)
- Right-hand limit as \( x \) approaches 1 from the right: \( \lim_{x \to 1^+} [x] = 1 \)
- Since the left-hand limit and right-hand limit do not equal each other, \( f(x) \) is discontinuous at \( x = 1 \).
3. Similarly, we can check other integers (e.g., \( x = 2, 3, \ldots \)) and find that the function is discontinuous at all integer values, not just at \( x = 0 \).

**Conclusion for Statement I:** This statement is incorrect because the function is discontinuous at all integers, not just at \( x = 0 \).

### Step 2: Analyze the second statement
**Statement II:** The function \( f(x) = \sin |x| \) is continuous for all \( x \in \mathbb{R} \).

1. The sine function, \( \sin(x) \), is continuous for all real numbers.
2. The absolute value function, \( |x| \), is also continuous for all real numbers.
3. The composition of continuous functions is continuous. Therefore, since both \( \sin(x) \) and \( |x| \) are continuous, their composition \( \sin |x| \) is continuous for all \( x \in \mathbb{R} \).

**Conclusion for Statement II:** This statement is correct.

### Final Conclusion
- **Statement I** is incorrect.
- **Statement II** is correct.

Thus, the answer is that only the second statement is correct.