Consider the following statements in respect of a function f(x)
I. f(x) continuous at x = a. if `underset(xrarra)limf(x)` exists
II. If f(x) is continuous at a point, then `(1)/(f(x))` is also continuous at the point.
Which of the above statements is/are correct?

Options

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

Answer

To solve the problem, we need to analyze the two statements regarding the function \( f(x) \).

### Step 1: Analyze Statement I
**Statement I:** \( f(x) \) is continuous at \( x = a \) if \( \lim_{x \to a} f(x) \) exists.

**Solution:**
For a function \( f(x) \) to be continuous at a point \( x = a \), three conditions must be satisfied:
1. \( f(a) \) must be defined (i.e., \( f(a) \) exists).
2. The limit \( \lim_{x \to a} f(x) \) must exist.
3. The limit must equal the function value at that point: \( \lim_{x \to a} f(x) = f(a) \).

Thus, while the existence of the limit is necessary for continuity, it is not sufficient on its own. Therefore, Statement I is **incorrect**.

### Step 2: Analyze Statement II
**Statement II:** If \( f(x) \) is continuous at a point, then \( \frac{1}{f(x)} \) is also continuous at that point.

**Solution:**
For \( \frac{1}{f(x)} \) to be continuous at a point \( x = b \), \( f(b) \) must be non-zero (i.e., \( f(b) \neq 0 \)). If \( f(b) = 0 \), then \( \frac{1}{f(b)} \) is undefined, which means \( \frac{1}{f(x)} \) cannot be continuous at that point.

Thus, Statement II is also **incorrect** because continuity of \( f(x) \) does not guarantee that \( \frac{1}{f(x)} \) is continuous unless \( f(b) \neq 0 \).

### Conclusion:
Both statements are incorrect. Therefore, the answer is that neither statement I nor statement II is correct.

### Final Answer:
**Neither statement I nor statement II is correct.**

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