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

Explanation

To determine which statement about the function \( f(x) = |x - 3| \) is not correct, we will analyze the properties of the function step by step.

### Step 1: Understanding the Function
The function \( f(x) = |x - 3| \) represents the absolute value of \( x - 3 \). This means that the function measures the distance of \( x \) from 3 on the number line.

### Step 2: Checking Continuity
A function is continuous at a point if the limit as \( x \) approaches that point equals the function's value at that point. We need to check continuity at \( x = 3 \).

1. **Calculate \( f(3) \)**:
\[ f(3) = |3 - 3| = |0| = 0 \]

2. **Check the left-hand limit as \( x \) approaches 3**:
\[ \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} |x - 3| = |3 - 3| = 0 \]

3. **Check the right-hand limit as \( x \) approaches 3**:
\[ \lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} |x - 3| = |3 - 3| = 0 \]

Since both the left-hand limit and right-hand limit equal \( f(3) \), we conclude that \( f(x) \) is continuous at \( x = 3 \).

### Step 3: Checking Differentiability
Next, we check if the function is differentiable at \( x = 3 \).

1. **Left-hand derivative at \( x = 3 \)**:
\[ f'(3^-) = \lim_{h \to 0} \frac{f(3) - f(3 - h)}{h} = \lim_{h \to 0} \frac{0 - |3 - (3 - h)|}{h} = \lim_{h \to 0} \frac{-|h|}{h} \]
For \( h > 0 \), this limit approaches -1, and for \( h < 0 \), it approaches 1. Thus, the left-hand derivative is -1.

2. **Right-hand derivative at \( x = 3 \)**:
\[ f'(3^+) = \lim_{h \to 0} \frac{f(3 + h) - f(3)}{h} = \lim_{h \to 0} \frac{|(3 + h) - 3| - 0}{h} = \lim_{h \to 0} \frac{|h|}{h} \]
For \( h > 0 \), this limit approaches 1, and for \( h < 0 \), it approaches -1. Thus, the right-hand derivative is 1.

Since the left-hand and right-hand derivatives are not equal, \( f(x) \) is not differentiable at \( x = 3 \).

### Step 4: Conclusion
Based on our analysis:
- The function is continuous at \( x = 3 \).
- The function is not differentiable at \( x = 3 \).

Therefore, the statement that is not correct is: "The function is not continuous at \( x = 3 \)."

### Final Answer
The correct option is: The function is not continuous at \( x = 3 \).

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