The domain of definition of the functions `f(x) = log_(e)(x-[x])`, is

Options

A. `R`
B. `R-Z`
C. `(0, oo)`
D. none of these

Explanation

To find the domain of the function \( f(x) = \log_e(x - [x]) \), where \([x]\) is the greatest integer function (or floor function), we need to analyze the expression inside the logarithm.

### Step 1: Understanding the Function
The function \( f(x) = \log_e(x - [x]) \) involves the logarithm of the expression \( x - [x] \). The term \( x - [x] \) represents the fractional part of \( x \), denoted as \( \{x\} \).

### Step 2: Conditions for the Logarithm
The logarithm function \( \log_e(a) \) is defined only when \( a > 0 \). Therefore, we need to ensure that:
\[ x - [x] > 0 \]

### Step 3: Analyzing the Fractional Part
The fractional part \( \{x\} = x - [x] \) is defined as the non-integer part of \( x \). It lies in the interval:
\[ 0 \leq \{x\} < 1 \]
Thus, \( \{x\} > 0 \) when \( x \) is not an integer.

### Step 4: Conclusion on the Domain
From the analysis, we conclude that \( x - [x] > 0 \) when \( x \) is not an integer. Therefore, the domain of the function \( f(x) \) excludes all integer values.

Thus, the domain of the function \( f(x) = \log_e(x - [x]) \) is:
\[ \text{Domain: } x \in \mathbb{R} \setminus \mathbb{Z} \]
This means \( x \) can be any real number except integers.