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, we need to analyze each of the four statements regarding continuity and differentiability of functions. Let's go through them step by step.

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

**Analysis:**
- A function can be continuous at a point but not differentiable there. For example, a function with a cusp (sharp point) is continuous but not differentiable at that point.
- Therefore, this statement is **incorrect**.

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

**Analysis:**
- If a function is differentiable at a point, it must also be continuous at that point. This is a fundamental property of differentiable functions.
- Therefore, this statement is **correct**.

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

**Analysis:**
- For a function to be continuous at a point, the left-hand limit, right-hand limit, and the function's value at that point must all be equal. The existence of a limit alone does not guarantee continuity.
- For example, consider the greatest integer function, which has limits that exist but is not continuous at integer points.
- Therefore, this statement is **incorrect**.

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

**Analysis:**
- A tangent can exist at a point even if the function is not differentiable there. For instance, a function with a sharp point can have a tangent line at that point but is not differentiable.
- Therefore, this statement is **incorrect**.

### Conclusion
After analyzing all four statements, we find that only **Statement 2** is correct.

### Final Answer
The correct statement is: **Differentiability at \( x = a \) is sufficient for continuity at \( x = a \)** (Statement 2).

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