The function `f(x)=x+|x|` is continuous for

Options

A. only `x gt 0`
B. ` x in (-oo, oo)- {0}`
C. ` x in (-oo, oo)`
D. no values of x

Answer

To determine the continuity of the function \( f(x) = x + |x| \), we need to analyze the function based on the definition of continuity and the behavior of the absolute value function.

### Step 1: Simplify the function based on the definition of absolute value
The absolute value function \( |x| \) can be expressed in piecewise form:
- If \( x \geq 0 \), then \( |x| = x \)
- If \( x < 0 \), then \( |x| = -x \)

Thus, we can rewrite \( f(x) \) as:
- For \( x \geq 0 \):
\[ f(x) = x + |x| = x + x = 2x \]
- For \( x < 0 \):
\[ f(x) = x + |x| = x - x = 0 \]

### Step 2: Define the piecewise function
Now we can define the function \( f(x) \) as:
\[ f(x) = \begin{cases} 0 & \text{if } x < 0 \\ 2x & \text{if } x \geq 0 \end{cases} \]

### Step 3: Check continuity at \( x = 0 \)
To check the continuity of \( f(x) \) at \( x = 0 \), we need to find the left-hand limit (LHL), the right-hand limit (RHL), and the value of the function at that point.

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

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

3. **Value of the function at \( x = 0 \)**:
\[ f(0) = 2 \cdot 0 = 0 \]

### Step 4: Compare limits and function value
Now we compare the limits and the function value:
- LHL = 0
- RHL = 0
- \( f(0) = 0 \)

Since LHL = RHL = \( f(0) \), the function \( f(x) \) is continuous at \( x = 0 \).

### Step 5: Conclusion about continuity
Since \( f(x) \) is defined for all \( x \) (both negative and positive) and is continuous at \( x = 0 \), we conclude that the function \( f(x) = x + |x| \) is continuous for all \( x \in \mathbb{R} \).

### Final Answer
The function \( f(x) = x + |x| \) is continuous for \( x \in (-\infty, \infty) \).