The function `F:RtoR` given by f(x)=-|x-1| is

Options

A. continuous well as differentiable at x=1
B. not continuous but differentiable at x=1
C. continuous but not differentiable at x=1
D. neither continuous nor differentiable at x=1.

Answer

To analyze the function \( f(x) = -|x - 1| \), we will determine its continuity and differentiability.

### Step 1: Understand the function
The function \( f(x) = -|x - 1| \) can be expressed in piecewise form based on the definition of the absolute value.

### Step 2: Break down the absolute value
The absolute value function \( |x - 1| \) can be defined as:
- \( x - 1 \) when \( x \geq 1 \)
- \( -(x - 1) = -x + 1 \) when \( x < 1 \)

Thus, we can rewrite \( f(x) \) as:
\[ f(x) = \begin{cases} -(x - 1) = -x + 1 & \text{if } x \geq 1 \\ -(-x + 1) = x - 1 & \text{if } x < 1 \end{cases} \]

### Step 3: Write the piecewise function
Now, we can summarize the piecewise function:
\[ f(x) = \begin{cases} -x + 1 & \text{if } x \geq 1 \\ x - 1 & \text{if } x < 1 \end{cases} \]

### Step 4: Check for continuity at \( x = 1 \)
To check continuity at \( x = 1 \), we need to evaluate:
1. \( f(1) \)
2. \( \lim_{x \to 1^-} f(x) \)
3. \( \lim_{x \to 1^+} f(x) \)

Calculating these:
- \( f(1) = -|1 - 1| = 0 \)
- \( \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x - 1) = 1 - 1 = 0 \)
- \( \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (-x + 1) = -1 + 1 = 0 \)

Since \( f(1) = \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = 0 \), the function is continuous at \( x = 1 \).

### Step 5: Check for differentiability at \( x = 1 \)
To check differentiability, we need to find the left-hand derivative and the right-hand derivative at \( x = 1 \).

1. **Left-hand derivative**:
\[ f'(x) = \frac{d}{dx}(x - 1) = 1 \quad \text{for } x < 1 \]
Thus, \( \lim_{x \to 1^-} f'(x) = 1 \).

2. **Right-hand derivative**:
\[ f'(x) = \frac{d}{dx}(-x + 1) = -1 \quad \text{for } x \geq 1 \]
Thus, \( \lim_{x \to 1^+} f'(x) = -1 \).

Since the left-hand derivative \( (1) \) is not equal to the right-hand derivative \( (-1) \), the function is not differentiable at \( x = 1 \).

### Conclusion
The function \( f(x) = -|x - 1| \) is continuous at \( x = 1 \) but not differentiable at that point.

### Final Answer
The function \( f(x) = -|x - 1| \) is continuous but not differentiable at \( x = 1 \).

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