If `f(x)= sin x`, then f is

Options

A. discontinuous for all `x in R`
B. continuous for all `x in R^(+)`
C. continuous for all ` x in R^(-)`
D. continuous for all `x in R`

Answer

To determine the continuity of the function \( f(x) = \sin x \), we will follow these steps:

### Step 1: Understand the Definition of Continuity
A function \( f(x) \) is continuous at a point \( c \) if the following three conditions are satisfied:
1. \( f(c) \) is defined.
2. \( \lim_{x \to c} f(x) \) exists.
3. \( \lim_{x \to c} f(x) = f(c) \).

### Step 2: Identify the Domain of \( f(x) = \sin x \)
The sine function is defined for all real numbers. Therefore, the domain of \( f(x) = \sin x \) is \( x \in \mathbb{R} \).

### Step 3: Check Continuity at Any Point \( c \)
Let \( c \) be any real number. We need to check the three conditions for continuity:

1. **Is \( f(c) \) defined?**
- Since \( f(c) = \sin c \) is defined for all \( c \in \mathbb{R} \), this condition is satisfied.

2. **Does \( \lim_{x \to c} f(x) \) exist?**
- The limit \( \lim_{x \to c} \sin x \) exists because sine is a well-defined function for all \( x \).

3. **Is \( \lim_{x \to c} f(x) = f(c) \)?**
- We have \( \lim_{x \to c} \sin x = \sin c \), which matches \( f(c) \).

### Step 4: Conclusion
Since all three conditions for continuity are satisfied for any point \( c \in \mathbb{R} \), we conclude that the function \( f(x) = \sin x \) is continuous everywhere on its domain.

### Final Answer
Thus, \( f(x) = \sin x \) is continuous for all \( x \in \mathbb{R} \).

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