If `(x) = |x| + |x - 1|,` than :

Options

A. f(x) is continous at x = 0 and at x = 1
B. f(x) is continous at x = -0 and discontinuous at a x = 1
C. f (x) is discontinous at x = 0 and continuous at x = 1
D. None of these

Answer

To solve the problem, we need to analyze the function \( f(x) = |x| + |x - 1| \) and determine where it is continuous. We will find the critical points, evaluate the function in different intervals, and check the limits at those critical points.

### Step-by-Step Solution:

1. **Identify Critical Points**:
The critical points occur where the expressions inside the absolute values change signs. For \( |x| \), the critical point is \( x = 0 \) and for \( |x - 1| \), the critical point is \( x = 1 \). Thus, the critical points are \( x = 0 \) and \( x = 1 \).

**Hint**: Look for points where the expression inside the absolute value equals zero.

2. **Define Intervals**:
We will analyze the function in three intervals based on the critical points:
- Interval 1: \( x < 0 \)
- Interval 2: \( 0 \leq x < 1 \)
- Interval 3: \( x \geq 1 \)

**Hint**: Break the real line into intervals based on the critical points found.

3. **Evaluate the Function in Each Interval**:
- **For \( x < 0 \)**:
\[ f(x) = -x + -(x - 1) = -x - x + 1 = -2x + 1 \]
- **For \( 0 \leq x < 1 \)**:
\[ f(x) = x + -(x - 1) = x - x + 1 = 1 \]
- **For \( x \geq 1 \)**:
\[ f(x) = x + (x - 1) = x + x - 1 = 2x - 1 \]

**Hint**: Substitute values from each interval into the function to simplify.

4. **Check Continuity at Critical Points**:
We need to check the limits as \( x \) approaches the critical points and compare them with the function values at those points.

- **At \( x = 0 \)**:
\[ \lim_{x \to 0^-} f(x) = -2(0) + 1 = 1 \]
\[ f(0) = 1 \]
\[ \lim_{x \to 0^+} f(x) = 1 \]
Since \( \lim_{x \to 0} f(x) = f(0) \), \( f(x) \) is continuous at \( x = 0 \).

- **At \( x = 1 \)**:
\[ \lim_{x \to 1^-} f(x) = 1 \]
\[ f(1) = 1 \]
\[ \lim_{x \to 1^+} f(x) = 2(1) - 1 = 1 \]
Since \( \lim_{x \to 1} f(x) = f(1) \), \( f(x) \) is continuous at \( x = 1 \).

**Hint**: Use the definition of continuity: a function is continuous at a point if the limit as you approach the point equals the function value at that point.

5. **Conclusion**:
Since \( f(x) \) is continuous at both critical points \( x = 0 \) and \( x = 1 \), we conclude that the function is continuous at these points.

**Final Statement**: The function \( f(x) = |x| + |x - 1| \) is continuous at \( x = 0 \) and \( x = 1 \).