Let f(x) and g(x) be two equal real function such that `f(x)=(x)/(|x|) g(x), x ne 0`
If g(0)=g'(0)=0 and f(x) is continuous at x=0, then f'(0) is

Options

A. 0
B. 1
C. `-1`
D. non-existent

Explanation

To solve the problem, we need to analyze the functions \( f(x) \) and \( g(x) \) given the conditions stated in the question.

### Step-by-Step Solution:

1. **Understanding the Functions**:
We have:
\[ f(x) = \frac{x}{|x|} g(x) \quad \text{for } x \neq 0 \]
This means:
- For \( x > 0 \), \( f(x) = g(x) \)
- For \( x < 0 \), \( f(x) = -g(x) \)

2. **Continuity at \( x = 0 \)**:
We know that \( f(x) \) is continuous at \( x = 0 \). Therefore, we need to check the left-hand limit (LHL) and right-hand limit (RHL) as \( x \) approaches 0.

- **Left-Hand Limit**:
\[ \text{LHL} = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} -g(x) = -g(0) = -0 = 0 \]

- **Right-Hand Limit**:
\[ \text{RHL} = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} g(x) = g(0) = 0 \]

Since both limits are equal, we have:
\[ \lim_{x \to 0} f(x) = 0 \]
Thus, \( f(0) = 0 \) ensures continuity at \( x = 0 \).

3. **Finding the Derivative \( f'(0) \)**:
To find \( f'(0) \), we will calculate the left-hand derivative (LHD) and right-hand derivative (RHD) at \( x = 0 \).

- **Left-Hand Derivative**:
\[ \text{LHD} = \lim_{h \to 0^-} \frac{f(0) - f(-h)}{-h} = \lim_{h \to 0^-} \frac{0 - (-g(h))}{-h} = \lim_{h \to 0^-} \frac{g(h)}{h} \]
Since \( g(0) = 0 \), we can apply L'Hôpital's Rule:
\[ \text{LHD} = \lim_{h \to 0^-} \frac{g(h)}{h} = g'(0) \]

- **Right-Hand Derivative**:
\[ \text{RHD} = \lim_{h \to 0^+} \frac{f(0) - f(h)}{h} = \lim_{h \to 0^+} \frac{0 - g(h)}{h} = -\lim_{h \to 0^+} \frac{g(h)}{h} \]
Again, applying L'Hôpital's Rule:
\[ \text{RHD} = -g'(0) \]

4. **Setting LHD equal to RHD**:
For \( f'(0) \) to exist, we need:
\[ g'(0) = -g'(0) \]
This implies:
\[ 2g'(0) = 0 \implies g'(0) = 0 \]

5. **Conclusion**:
Since both the left-hand and right-hand derivatives equal 0, we conclude that:
\[ f'(0) = 0 \]

### Final Answer:
Thus, the value of \( f'(0) \) is \( \boxed{0} \).