The function `f(x)={x} sin (pi[x])`, where [.] denotes the greatest integer function and {.} is the fractional part function, is discontinuous at

Options

A. all x
B. all integer points
C. no x
D. x which is not an integer

Answer

To analyze the function \( f(x) = \{x\} \sin(\pi [x]) \), where \([x]\) is the greatest integer function and \(\{x\}\) is the fractional part function, we will determine the points of discontinuity.

### Step-by-step solution:

1. **Understanding the Components of the Function**:
- 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 \(x\), which is always in the range \([0, 1)\).

2. **Behavior of \(\sin(\pi [x])\)**:
- The sine function \(\sin(\pi n)\) is equal to 0 for any integer \(n\). Therefore, \(\sin(\pi [x]) = 0\) whenever \([x]\) is an integer.
- This means that for all integer values of \(x\), \(f(x) = \{x\} \cdot 0 = 0\).

3. **Evaluating the Function at Non-Integer Points**:
- For non-integer values of \(x\), \([x]\) is an integer \(n\) and \(\{x\} = x - n\). Thus, \(f(x) = (x - n) \sin(\pi n)\).
- Since \(\sin(\pi n) = 0\) for any integer \(n\), it follows that \(f(x) = (x - n) \cdot 0 = 0\) for all non-integer \(x\) as well.

4. **Checking for Discontinuity**:
- To check for discontinuity, we need to evaluate the limits as \(x\) approaches any integer \(n\):
- As \(x\) approaches \(n\) from the left (\(x \to n^{-}\)), \(\{x\} \to 1\) and \(\sin(\pi [x]) = \sin(\pi (n-1)) = 0\). Thus, \(f(n^{-}) = 1 \cdot 0 = 0\).
- As \(x\) approaches \(n\) from the right (\(x \to n^{+}\)), \(\{x\} \to 0\) and \(\sin(\pi [x]) = \sin(\pi n) = 0\). Thus, \(f(n^{+}) = 0 \cdot 0 = 0\).
- Therefore, \(f(n^{-}) = f(n^{+}) = 0\), and \(f(n) = 0\).

5. **Conclusion**:
- Since \(f(x)\) approaches the same value (0) from both sides of every integer \(n\), and \(f(n) = 0\), the function is continuous at all points.
- Hence, the function \(f(x)\) is discontinuous at **no points**.