Consider the following statement in respect of a function f(x):
1. f(x) is continuous at x =a iff `lim_(xtoa) f(x)` exists.
2. If f(x) is continuous at a point, then `(1)/(f(x))` is also continuous at that point.
Which of the above, statements is/are corrent?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To analyze the given statements about the function \( f(x) \), we will evaluate each statement step by step.

### Step 1: Evaluate the first statement
The first statement is:
1. \( f(x) \) is continuous at \( x = a \) iff \( \lim_{x \to a} f(x) \) exists.

**Explanation:**
A function \( f(x) \) is said to be continuous at a point \( x = a \) if the following three conditions are satisfied:
- \( f(a) \) is defined.
- \( \lim_{x \to a} f(x) \) exists.
- \( \lim_{x \to a} f(x) = f(a) \).

The statement only mentions that \( \lim_{x \to a} f(x) \) exists, which is not sufficient for continuity because it does not guarantee that \( f(a) \) is defined or that the limit equals \( f(a) \). Therefore, the first statement is **false**.

### Step 2: Evaluate the second statement
The second statement is:
2. If \( f(x) \) is continuous at a point, then \( \frac{1}{f(x)} \) is also continuous at that point.

**Explanation:**
For \( \frac{1}{f(x)} \) to be continuous at \( x = a \), \( f(x) \) must be continuous at \( x = a \) and also \( f(a) \) must not be equal to zero (since division by zero is undefined). If \( f(a) = 0 \), then \( \frac{1}{f(x)} \) is not defined at that point, and thus it cannot be continuous. Therefore, the second statement is also **false**.

### Conclusion
Both statements are incorrect. Hence, the answer is that neither statement 1 nor statement 2 is correct.

### Final Answer
Both statements are false.

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