Find the domain of the function `f(x)=log_(e)(x-[x])`, where `[.]` denotes the greatest integer function.

Answer

To find the domain of the function \( f(x) = \log_e(x - [x]) \), where \([x]\) denotes the greatest integer function (also known as the floor function), we need to determine the values of \( x \) for which the function is defined.

### Step-by-step Solution:

1. **Understanding the Function**:
The function \( f(x) = \log_e(x - [x]) \) is defined when the argument of the logarithm, \( x - [x] \), is greater than 0. This is because the logarithm is only defined for positive real numbers.

2. **Analyzing \( x - [x] \)**:
The expression \( x - [x] \) represents the fractional part of \( x \). For any real number \( x \), \( [x] \) is the greatest integer less than or equal to \( x \). Therefore, \( x - [x] \) gives us the decimal part of \( x \).

3. **Finding the Condition for Positivity**:
We need to find when \( x - [x] > 0 \). This condition holds true when \( x \) is not an integer. If \( x \) is an integer, then \( x - [x] = 0 \), which is not allowed in the logarithm.

4. **Identifying the Domain**:
The values of \( x \) that make \( f(x) \) undefined are precisely the integers. Thus, the domain of \( f(x) \) consists of all real numbers except the integers.

5. **Expressing the Domain**:
We can express the domain in set notation as:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \mathbb{Z} \]
where \( \mathbb{R} \) is the set of all real numbers and \( \mathbb{Z} \) is the set of all integers.

### Final Answer:
The domain of the function \( f(x) = \log_e(x - [x]) \) is \( \mathbb{R} \setminus \mathbb{Z} \).

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