Which of the following is a correct statement ?

Options

A. Continuity at x = a is sufficient for differentiability at x = a
B. Differentiability at x = a is sufficient for continuity at x = a
C. Existence of limit at x = a is sufficient for continuity at x = a
D. Differentiability at x = a is necessary for existence of tangent at x = a

Answer

To solve the question of which statement is correct regarding continuity and differentiability, we will analyze each option step by step.

### Step 1: Analyze Option 1
**Statement:** Continuity at \( x = a \) is sufficient for differentiability at \( x = a \).

- **Example:** Consider the function \( f(x) = |x| \).
- **At \( x = 0 \):** The function is continuous because \( \lim_{x \to 0} f(x) = f(0) = 0 \).
- **Differentiability:** However, the derivative \( f'(0) \) does not exist because the left-hand derivative (approaching from the left) is -1 and the right-hand derivative (approaching from the right) is +1. Since the derivatives do not match, \( f(x) \) is not differentiable at \( x = 0 \).

**Conclusion:** This option is incorrect.

### Step 2: Analyze Option 2
**Statement:** Differentiability at \( x = a \) is sufficient for continuity at \( x = a \).

- **Definition:** If a function is differentiable at a point, it must also be continuous at that point.
- **Reasoning:** The existence of the derivative implies that the limit defining the derivative exists, which requires the function to be continuous at that point.

**Conclusion:** This option is correct.

### Step 3: Analyze Option 3
**Statement:** The existence of the limit at \( x = a \) is sufficient for continuity at \( x = a \).

- **Example:** Consider the function defined as:
\[ f(x) = \begin{cases} 1 & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \]
- **Limit at \( x = 0 \):** The limit exists as \( x \) approaches 0, which is 1.
- **Continuity:** However, \( f(0) = 0 \), which is not equal to the limit. Therefore, the function is not continuous at \( x = 0 \).

**Conclusion:** This option is incorrect.

### Step 4: Analyze Option 4
**Statement:** Differentiability at \( x = a \) is necessary for the existence of a tangent at \( x = a \).

- **Example:** Consider the function \( f(x) = |x| \) again.
- **At \( x = 0 \):** The function has a tangent line (the line \( y = 0 \)), but it is not differentiable at that point due to the sharp corner.

**Conclusion:** This option is incorrect.

### Final Conclusion
After analyzing all options, the correct statement is:

**Option 2:** Differentiability at \( x = a \) is sufficient for continuity at \( x = a \).