The number of points at which the function `f(x) = 1/(log|x|)` is discontinuous is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine the number of points at which the function \( f(x) = \frac{1}{\log |x|} \) is discontinuous, we will follow these steps:

### Step 1: Identify where the function is undefined
The function \( f(x) \) is undefined when the denominator \( \log |x| \) is equal to zero. Therefore, we need to solve the equation:
\[ \log |x| = 0 \]

### Step 2: Solve for \( |x| \)
The logarithm is zero when its argument is equal to 1. Thus, we have:
\[ |x| = 1 \]
This gives us two cases:
1. \( x = 1 \)
2. \( x = -1 \)

### Step 3: Identify points of discontinuity
Next, we need to check if there are any other points where the function could be discontinuous. The function \( f(x) \) is also undefined at \( x = 0 \) because \( \log |x| \) is undefined at \( x = 0 \).

### Step 4: List all points of discontinuity
From the above analysis, we have identified the following points where \( f(x) \) is discontinuous:
1. \( x = 1 \)
2. \( x = -1 \)
3. \( x = 0 \)

### Step 5: Count the points of discontinuity
Thus, the total number of points at which the function \( f(x) \) is discontinuous is:
\[ \text{Total points} = 3 \]

### Final Answer
The number of points at which the function \( f(x) = \frac{1}{\log |x|} \) is discontinuous is **3**.

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