For the function `f(x)=|x-3|`, which of the following is not correct?

Options

A. The function is not continuous at `x=-3`
B. The function is continuous at `x=3`
C. The function is differentiable at `x=0`
D. The function is differentiable at `x=-3`

Answer

To determine which statement about the function \( f(x) = |x - 3| \) is not correct, we will analyze the function's continuity and differentiability at specific points.

### Step 1: Understand the function
The function \( f(x) = |x - 3| \) can be expressed in piecewise form:
- For \( x < 3 \), \( f(x) = 3 - x \)
- For \( x \geq 3 \), \( f(x) = x - 3 \)

### Step 2: Check continuity at \( x = 3 \)
To check continuity at \( x = 3 \), we need to find the left-hand limit (LHL), right-hand limit (RHL), and the function value at \( x = 3 \).

- **LHL at \( x = 3 \)**:
\[ \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (3 - x) = 3 - 3 = 0 \]

- **RHL at \( x = 3 \)**:
\[ \lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} (x - 3) = 3 - 3 = 0 \]

- **Function value at \( x = 3 \)**:
\[ f(3) = |3 - 3| = 0 \]

Since LHL = RHL = \( f(3) = 0 \), the function is continuous at \( x = 3 \).

### Step 3: Check differentiability at \( x = 3 \)
To check differentiability, we need to find the derivative from both sides.

- **Derivative for \( x < 3 \)**:
\[ f'(x) = -1 \quad \text{(since } f(x) = 3 - x \text{)} \]

- **Derivative for \( x \geq 3 \)**:
\[ f'(x) = 1 \quad \text{(since } f(x) = x - 3 \text{)} \]

At \( x = 3 \), the left-hand derivative is -1 and the right-hand derivative is 1. Since these two values are not equal, \( f(x) \) is not differentiable at \( x = 3 \).

### Step 4: Check differentiability at \( x = 0 \)
For \( x = 0 \):
- Since \( 0 < 3 \), we use the expression \( f(x) = 3 - x \).
- The derivative \( f'(0) = -1 \), which exists. Thus, the function is differentiable at \( x = 0 \).

### Step 5: Check continuity at \( x = -3 \)
For \( x = -3 \):
- Since \( -3 < 3 \), we use the expression \( f(x) = 3 - x \).
- The function is continuous everywhere, including at \( x = -3 \).

### Conclusion
Now we can evaluate the options:
1. The function is not continuous at \( x = -3 \) - **Incorrect** (the function is continuous everywhere).
2. The function is continuous at \( x = 3 \) - **Correct**.
3. The function is differentiable at \( x = 0 \) - **Correct**.
4. The function is differentiable at \( x = -3 \) - **Correct**.

Thus, the statement that is **not correct** is option 1.