If `F(x)=(sinpi[x])/({x})` then `F(x)` is (where {.} denotes fractional part function and [.] denotes greatest integer function and `sgn(x)` is a signum function)

Options

A. periodic with fundamental period 1
B. even
C. range is singleton
D. indentical to `sgn(sgn({x})/(sqrt({x})))-1`

Explanation

To solve the problem, we need to analyze the function \( F(x) = \frac{\sin(\pi [x])}{\{x\}} \), where \([x]\) is the greatest integer function (floor function) and \(\{x\}\) is the fractional part function.

### Step-by-Step Solution:

1. **Understanding the Components**:
- The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- The fractional part function \(\{x\} = x - [x]\) gives the non-integer part of \(x\).
- The sine function \(\sin(\pi n) = 0\) for any integer \(n\).

2. **Evaluating \(F(x)\) at Integer Values**:
- If \(x\) is an integer, then \([x] = x\) and \(\{x\} = 0\).
- Thus, \(F(x) = \frac{\sin(\pi x)}{0}\) which is undefined.

3. **Evaluating \(F(x)\) at Non-Integer Values**:
- For non-integer values of \(x\), \([x]\) is the integer part of \(x\) and \(\{x\}\) is positive (since it is the non-integer part).
- Therefore, \(F(x) = \frac{\sin(\pi [x])}{\{x\}}\).

4. **Analyzing the Sine Function**:
- For \(x\) in the interval \([n, n+1)\) where \(n\) is an integer, \([x] = n\).
- Thus, \(F(x) = \frac{\sin(\pi n)}{\{x\}} = \frac{0}{\{x\}} = 0\) since \(\sin(\pi n) = 0\).

5. **Conclusion on the Function**:
- Since \(F(x) = 0\) for all non-integer \(x\) and is undefined for integer \(x\), the function \(F(x)\) is essentially zero for all non-integers.
- The **domain** of \(F(x)\) is all real numbers except integers, i.e., \( \mathbb{R} \setminus \mathbb{Z} \).
- The **range** of \(F(x)\) is just \(\{0\}\) (singleton set).

6. **Identifying the Nature of \(F(x)\)**:
- \(F(x)\) is not periodic since it is constantly zero for non-integer values.
- \(F(x)\) is an even function because \(F(-x) = F(x) = 0\) for all non-integer \(x\).
- \(F(x)\) is not identical to any other function since it has a specific domain and range.

### Final Answer:
- The function \(F(x)\) is an **even function** and has a range of \(\{0\}\) (singleton).