The set of points of discontinuity of the function `f(x)=(1)/(log|x|),is`

Options

A. one point
B. two points
C. three points
D. infinite number of points

Answer

To find the points of discontinuity of the function \( f(x) = \frac{1}{\log |x|} \), we need to determine where the function is undefined or where it approaches infinity.

### Step-by-Step Solution:

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

2. **Determine when the denominator is zero**:
The function \( f(x) \) will be undefined when the denominator \( \log |x| = 0 \).

3. **Solve for when \( \log |x| = 0 \)**:
The logarithm is zero when its argument is 1:
\[ |x| = 1 \]
This gives us two cases:
\[ x = 1 \quad \text{and} \quad x = -1 \]

4. **Check for points where the function is undefined**:
The logarithm is also undefined for \( |x| \leq 0 \). Since \( \log |x| \) is undefined for \( x = 0 \), we must also consider this point.

5. **List the points of discontinuity**:
From the above analysis, the points of discontinuity are:
\[ x = -1, \quad x = 0, \quad x = 1 \]

### Conclusion:
The set of points of discontinuity of the function \( f(x) = \frac{1}{\log |x|} \) is:
\[ \{-1, 0, 1\} \]