Let `f:ArarrR`, where `A=R//(0)` is such that `f(x)=(x+|x|)/(x)`. On which one of the following sets is f(x) continuous?

Options

A. 1) `A`
B. 2) `B={x inR:xge0}`
C. 3) `C={x inR:xle0}`
D. 4) `D = R`

Answer

To determine the continuity of the function \( f(x) = \frac{x + |x|}{x} \) on the set \( A = \mathbb{R} \setminus \{0\} \), we will analyze the function in two parts: when \( x < 0 \) and when \( x > 0 \).

### Step 1: Analyze the function for \( x < 0 \)

For \( x < 0 \):
- The absolute value \( |x| = -x \).
- Therefore, the function becomes:
\[ f(x) = \frac{x + (-x)}{x} = \frac{0}{x} = 0 \]
- Thus, for all \( x < 0 \), \( f(x) = 0 \).

### Step 2: Analyze the function for \( x > 0 \)

For \( x > 0 \):
- The absolute value \( |x| = x \).
- Therefore, the function becomes:
\[ f(x) = \frac{x + x}{x} = \frac{2x}{x} = 2 \]
- Thus, for all \( x > 0 \), \( f(x) = 2 \).

### Step 3: Check continuity at \( x = 0 \)

To check continuity at \( x = 0 \), we need to evaluate the left-hand limit and the right-hand limit as \( x \) approaches 0.

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

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

Since the left-hand limit (0) does not equal the right-hand limit (2), we conclude that \( f(x) \) is discontinuous at \( x = 0 \).

### Step 4: Conclusion about continuity on the set \( A \)

The function \( f(x) \) is defined for all \( x \in \mathbb{R} \setminus \{0\} \) and is continuous at every point in this set. Therefore, we conclude that:

- The function \( f(x) \) is continuous on the set \( A = \mathbb{R} \setminus \{0\} \).

### Final Answer

The function \( f(x) \) is continuous on the set \( A = \mathbb{R} \setminus \{0\} \).

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