At the point x = 1, the function:
`f(x) = {{:(x^(3)-1, 1 lt x lt infty),(x-1, -infty lt x le 1):}` is:

Options

A. continuous and differentiable
B. continuous and not differentiable
C. discontinuous and differentiable
D. none of these

Answer

To determine the continuity and differentiability of the function \( f(x) \) at the point \( x = 1 \), we will follow these steps:

### Step 1: Define the Function
The function is defined as:
\[ f(x) = \begin{cases} x^3 - 1 & \text{for } 1 < x < \infty \\ x - 1 & \text{for } -\infty < x \leq 1 \end{cases} \]

### Step 2: Check Continuity at \( x = 1 \)
To check for continuity at \( x = 1 \), we need to verify that:
\[ \lim_{x \to 1^-} f(x) = f(1) = \lim_{x \to 1^+} f(x) \]

#### Step 2.1: Calculate \( f(1) \)
From the definition of the function, since \( 1 \leq 1 \), we use the second case:
\[ f(1) = 1 - 1 = 0 \]

#### Step 2.2: Calculate Left-Hand Limit
The left-hand limit as \( x \) approaches 1 is given by the second case of the function:
\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x - 1) = 1 - 1 = 0 \]

#### Step 2.3: Calculate Right-Hand Limit
The right-hand limit as \( x \) approaches 1 is given by the first case of the function:
\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x^3 - 1) = 1^3 - 1 = 0 \]

#### Step 2.4: Compare Limits and Function Value
Since:
\[ \lim_{x \to 1^-} f(x) = 0, \quad f(1) = 0, \quad \lim_{x \to 1^+} f(x) = 0 \]
All three values are equal, hence \( f(x) \) is continuous at \( x = 1 \).

### Step 3: Check Differentiability at \( x = 1 \)
To check for differentiability, we need to verify that the left-hand derivative equals the right-hand derivative at \( x = 1 \).

#### Step 3.1: Calculate Left-Hand Derivative
The left-hand derivative at \( x = 1 \) is calculated using the second case of the function:
\[ f'(x) = 1 \quad \text{for } x < 1 \]
Thus,
\[ \lim_{h \to 0^-} \frac{f(1 + h) - f(1)}{h} = 1 \]

#### Step 3.2: Calculate Right-Hand Derivative
The right-hand derivative at \( x = 1 \) is calculated using the first case of the function:
\[ f'(x) = 3x^2 \quad \text{for } x > 1 \]
Thus,
\[ \lim_{h \to 0^+} \frac{f(1 + h) - f(1)}{h} = \lim_{h \to 0^+} \frac{(1 + h)^3 - 1}{h} = \lim_{h \to 0^+} \frac{3h^2 + 3h + h^3}{h} = 3 \]

### Step 4: Compare Derivatives
Since:
\[ \text{Left-hand derivative} = 1 \quad \text{and} \quad \text{Right-hand derivative} = 3 \]
These two derivatives are not equal, hence \( f(x) \) is not differentiable at \( x = 1 \).

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

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