`f(x) ` is continuous at `x = 0` then which of the following are always true ?

Options

A. A. `underset(x rarr 0)(lim) f(x) = 0`
B. B. f(x) is non coninuous at x = 1
C. C. `g(x) = x^(2) f(x)` is continuous x = 0
D. D. `underset(x rarr 0^(+))(lim) (f(x) - f(0)) = 0`

Answer

To determine which statements are always true given that \( f(x) \) is continuous at \( x = 0 \), let's analyze each option step by step.

### Step 1: Understand the definition of continuity
A function \( f(x) \) is continuous at \( x = 0 \) if:
1. \( f(0) \) is defined.
2. \( \lim_{x \to 0} f(x) \) exists.
3. \( \lim_{x \to 0} f(x) = f(0) \).

### Step 2: Analyze each option

**Option A:** \( \lim_{x \to 0} f(x) = 0 \)

- This statement is not necessarily true. While \( f(x) \) is continuous at \( x = 0 \), it does not imply that the limit must equal zero. The limit could be any value that equals \( f(0) \). Therefore, this statement is **false**.

**Option B:** \( f(x) \) is non-continuous at \( x = 1 \)

- The continuity of \( f(x) \) at \( x = 0 \) does not provide any information about its behavior at \( x = 1 \). Thus, we cannot conclude anything about the continuity of \( f(x) \) at \( x = 1 \). Therefore, this statement is **false**.

**Option C:** \( g(x) = x^2 f(x) \) is continuous at \( x = 0 \)

- Since \( f(x) \) is continuous at \( x = 0 \) and \( x^2 \) is a polynomial (and hence continuous everywhere), the product of two continuous functions is also continuous. Therefore, \( g(x) = x^2 f(x) \) is continuous at \( x = 0 \). This statement is **true**.

**Option D:** \( \lim_{x \to 0^+} (f(x) - f(0)) = 0 \)

- For \( f(x) \) to be continuous at \( x = 0 \), it must hold that \( \lim_{x \to 0} f(x) = f(0) \). This implies that both the left-hand limit and the right-hand limit equal \( f(0) \). Hence, \( \lim_{x \to 0^+} (f(x) - f(0)) = 0 \) is indeed true. This statement is **true**.

### Conclusion
From the analysis, we conclude that the statements that are always true are:
- **C:** \( g(x) = x^2 f(x) \) is continuous at \( x = 0 \).
- **D:** \( \lim_{x \to 0^+} (f(x) - f(0)) = 0 \).

### Summary of Results
- **A:** False
- **B:** False
- **C:** True
- **D:** True