To solve the problem, we need to analyze the function \( f(x) = \frac{\sin(\pi [x])}{x} \), where \([x]\) denotes the greatest integer function (also known as the floor function).
### Step-by-Step Solution:
1. **Understanding the Greatest Integer Function**:
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For example:
- If \(x = 2.5\), then \([x] = 2\).
- If \(x = -1.3\), then \([x] = -2\).
2. **Analyzing the Function**:
The function can be rewritten as:
\[
f(x) = \frac{\sin(\pi [x])}{x}
\]
We need to determine the behavior of \(f(x)\) based on the values of \([x]\).
3. **Values of \(\sin(\pi [x])\)**:
- When \([x] = n\) (where \(n\) is an integer), \(\sin(\pi n) = 0\) for any integer \(n\).
- Therefore, for any integer \(x\), \(f(x) = \frac{0}{x} = 0\).
4. **Behavior Near Integer Values**:
- For \(x\) approaching an integer (but not equal to it), \([x]\) will be the integer just less than \(x\).
- For example, if \(x = 2.9\), then \([x] = 2\) and \(\sin(\pi [x]) = \sin(2\pi) = 0\).
- Hence, \(f(x) = \frac{0}{2.9} = 0\).
5. **Identifying the Domain**:
- The function is undefined at \(x = 0\) and at any integer \(x\) because we cannot divide by zero.
- Thus, the domain of \(f(x)\) is \( \mathbb{R} \setminus \{0, \text{integers}\} \).
6. **Periodicity**:
- The function \(f(x)\) is periodic with a fundamental period of 1 because \(\sin(\pi [x])\) will repeat its values every integer interval.
7. **Even Function**:
- To check if \(f(x)\) is an even function, we evaluate \(f(-x)\):
\[
f(-x) = \frac{\sin(\pi [-x])}{-x}
\]
Since \([-x] = -[x] - 1\) for non-integer \(x\), we find that \(\sin(\pi [-x]) = -\sin(\pi [x])\), leading to:
\[
f(-x) = -\frac{\sin(\pi [x])}{-x} = \frac{\sin(\pi [x])}{x} = f(x)
\]
Thus, \(f(x)\) is indeed an even function.
8. **Range**:
- The range of \(f(x)\) is \{0\} since \(f(x) = 0\) for all integer values of \(x\) and approaches 0 as \(x\) approaches any integer.
### Conclusion:
The function \(f(x)\) is periodic with a period of 1, is an even function, and its range is \{0\}.
