if f(x) =`x-[x], in R,` then `f^(')(1/2)` is equal to

Options

A. `3/2`
B. 1
C. 0
D. `-1`

Explanation

To solve the problem, we need to find the derivative of the function \( f(x) = x - [x] \) at the point \( x = \frac{1}{2} \), where \( [x] \) denotes the greatest integer function (also known as the floor function).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = x - [x] \) represents the fractional part of \( x \). For any real number \( x \), \( [x] \) is the greatest integer less than or equal to \( x \).

2. **Finding the Value of \( f(x) \) in the Interval**:
Since we are interested in \( x = \frac{1}{2} \), we note that \( \frac{1}{2} \) lies between 0 and 1. Therefore, in this interval:
\[ [x] = 0 \quad \text{for} \quad 0 \leq x < 1 \]
Thus, we can express \( f(x) \) as:
\[ f(x) = x - 0 = x \quad \text{for} \quad 0 \leq x < 1 \]

3. **Differentiating the Function**:
Now, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x) = 1 \]
This derivative holds for all \( x \) in the interval \( [0, 1) \).

4. **Evaluating the Derivative at \( x = \frac{1}{2} \)**:
Since \( \frac{1}{2} \) is in the interval \( [0, 1) \), we can directly substitute:
\[ f'\left(\frac{1}{2}\right) = 1 \]

### Final Answer:
Thus, \( f'(\frac{1}{2}) = 1 \).