Consider the following in respect of the function `f(x)=|x-3|`.
I. f(x) is continuous at x = 3
II. F(x) is differentiable at x = 0
Which of the above statement us/are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To determine the correctness of the statements regarding the function \( f(x) = |x - 3| \), we will analyze both statements step by step.

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

1. **Definition of Continuity**: A function is continuous at a point \( c \) if:
- \( f(c) \) is defined.
- \( \lim_{x \to c} f(x) \) exists.
- \( \lim_{x \to c} f(x) = f(c) \).

2. **Evaluate \( f(3) \)**:
\[ f(3) = |3 - 3| = |0| = 0. \]

3. **Evaluate the limit as \( x \) approaches 3**:
- For \( x < 3 \):
\[ f(x) = |x - 3| = 3 - x. \]
- For \( x > 3 \):
\[ f(x) = |x - 3| = x - 3. \]

4. **Calculate the left-hand limit**:
\[ \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (3 - x) = 3 - 3 = 0. \]

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

6. **Conclusion for continuity**:
\[ \lim_{x \to 3} f(x) = 0 = f(3). \]
Thus, \( f(x) \) is continuous at \( x = 3 \).

### Step 2: Check Differentiability at \( x = 0 \)

1. **Definition of Differentiability**: A function is differentiable at a point \( c \) if the derivative \( f'(c) \) exists, which requires that the left-hand and right-hand derivatives at that point are equal.

2. **Evaluate the derivative**:
- For \( x < 3 \) (including \( x = 0 \)):
\[ f(x) = 3 - x \quad \Rightarrow \quad f'(x) = -1. \]
- For \( x > 3 \):
\[ f(x) = x - 3 \quad \Rightarrow \quad f'(x) = 1. \]

3. **Evaluate the derivative at \( x = 0 \)**:
Since \( 0 < 3 \), we use the left side:
\[ f'(0) = -1. \]

4. **Conclusion for differentiability**:
Since the derivative exists and is defined at \( x = 0 \), \( f(x) \) is differentiable at \( x = 0 \).

### Final Conclusion

- Statement I: \( f(x) \) is continuous at \( x = 3 \) - **True**.
- Statement II: \( f(x) \) is differentiable at \( x = 0 \) - **True**.

Both statements are correct.

### Answer: Both statements are correct.

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