The set of numbers when the function `f(x)=(1)/(log|x|)` is discontinuous is (i) `{0}` (ii) `(1,-1)` (iii) `{0,1,-1}` (iv) none of these

Options

A. {0}
B. (1,-1)
C. {0,1,-1}
D. none of these

Answer

To determine the set of numbers where the function \( f(x) = \frac{1}{\log |x|} \) is discontinuous, we need to analyze the points where the function is undefined or where the logarithm function has issues.

### Step-by-Step Solution:

1. **Identify the function**:
We have \( f(x) = \frac{1}{\log |x|} \).

2. **Determine where \( \log |x| \) is undefined**:
The logarithm function \( \log |x| \) is undefined at \( x = 0 \) because the logarithm of zero is undefined. Thus, \( f(x) \) is undefined at \( x = 0 \).

3. **Find where \( \log |x| = 0 \)**:
The logarithm equals zero when its argument is 1. Therefore, we set:
\[ \log |x| = 0 \implies |x| = 1 \]
This gives us two points:
\[ x = 1 \quad \text{and} \quad x = -1 \]

4. **List the points of discontinuity**:
From the analysis, we find that \( f(x) \) is discontinuous at:
- \( x = 0 \) (where \( f(x) \) is undefined)
- \( x = 1 \) (where \( \log |x| = 0 \))
- \( x = -1 \) (where \( \log |x| = 0 \))

5. **Conclusion**:
Therefore, the set of points where the function \( f(x) \) is discontinuous is:
\[ \{ -1, 0, 1 \} \]

### Final Answer:
The correct option is (iii) \( \{0, 1, -1\} \).