If `f(x)=|x|+|x-1|`, then which one of the following is correct?

Options

A. f(x) is continuous at x=0 and x=1
B. f(x) is continuous at x=0 but not at x=1
C. f(x) is continuous at x=1 but not at x=0
D. f(x) is neither continuous at x=0 nor at x=1

Explanation

To solve the problem of determining the continuity of the function \( f(x) = |x| + |x - 1| \), we will analyze the function step by step.

### Step 1: Identify the points where the expression inside the absolute values changes sign.
The absolute value function \( |x| \) changes at \( x = 0 \) and \( |x - 1| \) changes at \( x = 1 \). Therefore, we need to consider the intervals defined by these points:
- \( x < 0 \)
- \( 0 \leq x < 1 \)
- \( x \geq 1 \)

### Step 2: Define the function piecewise based on the intervals.
1. For \( x < 0 \):
\[ f(x) = -x + -(x - 1) = -x - x + 1 = -2x + 1 \]
2. For \( 0 \leq x < 1 \):
\[ f(x) = x + -(x - 1) = x - x + 1 = 1 \]
3. For \( x \geq 1 \):
\[ f(x) = x + (x - 1) = x + x - 1 = 2x - 1 \]

### Step 3: Write the piecewise function.
Thus, we can express \( f(x) \) as:
\[ f(x) = \begin{cases} -2x + 1 & \text{if } x < 0 \\ 1 & \text{if } 0 \leq x < 1 \\ 2x - 1 & \text{if } x \geq 1 \end{cases} \]

### Step 4: Check continuity at the points \( x = 0 \) and \( x = 1 \).
1. **At \( x = 0 \)**:
- Left-hand limit: \( \lim_{x \to 0^-} f(x) = -2(0) + 1 = 1 \)
- Right-hand limit: \( \lim_{x \to 0^+} f(x) = 1 \)
- Function value: \( f(0) = 1 \)
- Since both limits and the function value are equal, \( f(x) \) is continuous at \( x = 0 \).

2. **At \( x = 1 \)**:
- Left-hand limit: \( \lim_{x \to 1^-} f(x) = 1 \)
- Right-hand limit: \( \lim_{x \to 1^+} f(x) = 2(1) - 1 = 1 \)
- Function value: \( f(1) = 1 \)
- Since both limits and the function value are equal, \( f(x) \) is continuous at \( x = 1 \).

### Step 5: Conclusion about continuity.
Since \( f(x) \) is continuous at both critical points \( x = 0 \) and \( x = 1 \), we conclude that the function is continuous for all \( x \in \mathbb{R} \).

### Final Answer:
The correct option is that \( f(x) \) is continuous for all \( x \in \mathbb{R} \).

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