If `f : R -> R` is defined by `f(x) = [2x] - 2[x]` for `x in R`, where [x] is the greatest integer not exceeding x, then the range of f is

Options

A. [0,1]
B. {0,1}
C. `(0,oo)`
D. `(-oo,0]`

Answer

To find the range of the function \( f : \mathbb{R} \to \mathbb{R} \) defined by

\[ f(x) = [2x] - 2[x] \]

where \([x]\) denotes the greatest integer not exceeding \(x\), we will analyze the function step by step.

### Step 1: Understanding the greatest integer function
Let \( x \) be expressed as \( x = n + f \), where \( n \) is an integer (i.e., \([x] = n\)) and \( f \) is the fractional part of \( x \) such that \( 0 \leq f < 1 \).

### Step 2: Calculate \( [2x] \)
Now, we calculate \( 2x \):

\[ 2x = 2(n + f) = 2n + 2f \]

The greatest integer function applied to \( 2x \) gives us:

\[ [2x] = [2n + 2f] \]

Since \( 2f \) lies in the interval \([0, 2)\), we can conclude:

- If \( 0 \leq 2f < 1 \), then \([2x] = 2n\)
- If \( 1 \leq 2f < 2 \), then \([2x] = 2n + 1\)

### Step 3: Evaluate \( f(x) \)
Now we substitute back into the function \( f(x) \):

1. **Case 1**: When \( 0 \leq 2f < 1 \):
\[ f(x) = [2x] - 2[x] = 2n - 2n = 0 \]

2. **Case 2**: When \( 1 \leq 2f < 2 \):
\[ f(x) = [2x] - 2[x] = (2n + 1) - 2n = 1 \]

### Step 4: Determine the range of \( f(x) \)
From the two cases analyzed, we see that:

- In the first case, \( f(x) = 0 \)
- In the second case, \( f(x) = 1 \)

Thus, the function \( f(x) \) can only take the values \( 0 \) and \( 1 \).

### Conclusion
The range of the function \( f \) is:

\[ \text{Range of } f = \{0, 1\} \]