Determine the values of x for which the following functions fails to be continuous or differentiable `f(x) = {{:((1-x)",",x lt 1),((1-x)(2-x)",",1 le x le 2),((3-x)",",x gt 2):}` justify your answer.

Answer

To determine the values of \( x \) for which the function

\[ f(x) = \begin{cases} 1 - x & \text{if } x < 1 \\ (1 - x)(2 - x) & \text{if } 1 \leq x \leq 2 \\ 3 - x & \text{if } x > 2 \end{cases} \]

fails to be continuous or differentiable, we will analyze the function at the points where the definition of the function changes, specifically at \( x = 1 \) and \( x = 2 \).

### Step 1: Check Continuity at \( x = 1 \)

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

2. **Right-hand limit** as \( x \) approaches 1:
\[ \lim_{x \to 1^+} f(x) = (1 - 1)(2 - 1) = 0 \]

3. **Value of the function** at \( x = 1 \):
\[ f(1) = (1 - 1)(2 - 1) = 0 \]

Since the left-hand limit, right-hand limit, and the value of the function at \( x = 1 \) are all equal, we conclude that \( f(x) \) is continuous at \( x = 1 \).

### Step 2: Check Continuity at \( x = 2 \)

1. **Left-hand limit** as \( x \) approaches 2:
\[ \lim_{x \to 2^-} f(x) = (2 - 1)(2 - 2) = 0 \]

2. **Right-hand limit** as \( x \) approaches 2:
\[ \lim_{x \to 2^+} f(x) = 3 - 2 = 1 \]

3. **Value of the function** at \( x = 2 \):
\[ f(2) = (1 - 2)(2 - 2) = 0 \]

Since the left-hand limit (0) is not equal to the right-hand limit (1), \( f(x) \) is not continuous at \( x = 2 \).

### Step 3: Check Differentiability at \( x = 1 \)

1. **Left-hand derivative** at \( x = 1 \):
\[ f'(x) = -1 \quad \text{for } x < 1 \]

2. **Right-hand derivative** at \( x = 1 \):
\[ f'(x) = 2x - 3 \quad \text{for } 1 \leq x \leq 2 \]
Evaluating at \( x = 1 \):
\[ f'(1) = 2(1) - 3 = -1 \]

Since the left-hand derivative and right-hand derivative at \( x = 1 \) are equal, \( f(x) \) is differentiable at \( x = 1 \).

### Step 4: Check Differentiability at \( x = 2 \)

1. **Left-hand derivative** at \( x = 2 \):
\[ f'(x) = 2x - 3 \quad \text{for } 1 \leq x \leq 2 \]
Evaluating at \( x = 2 \):
\[ f'(2) = 2(2) - 3 = 1 \]

2. **Right-hand derivative** at \( x = 2 \):
\[ f'(x) = -1 \quad \text{for } x > 2 \]

Since the left-hand derivative (1) is not equal to the right-hand derivative (-1), \( f(x) \) is not differentiable at \( x = 2 \).

### Conclusion

The function \( f(x) \) fails to be continuous at \( x = 2 \) and is also not differentiable at \( x = 2 \). Therefore, the final answer is:

**The function fails to be continuous and differentiable at \( x = 2 \).**

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