The function f(x)={x} , where [x] denotes the greatest integer function , is continuous at

Options

A. `-2`
B. 1
C. 4
D. 1.5

Answer

To determine the points at which the function \( f(x) = [x] \) (where \([x]\) denotes the greatest integer function) is continuous, we need to analyze the behavior of the function around integer and non-integer values.

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The greatest integer function \([x]\) returns the largest integer less than or equal to \(x\). For example:
- \([2.3] = 2\)
- \([2] = 2\)
- \([3.9] = 3\)
- \([-1.5] = -2\)

2. **Identifying Points of Discontinuity**:
The function \([x]\) is continuous at all non-integer points. However, it is discontinuous at integer points. This is because as \(x\) approaches an integer from the left, \([x]\) takes the value of the integer minus one, and as \(x\) approaches from the right, \([x]\) takes the value of the integer itself.

3. **Analyzing Given Points**:
We need to check the continuity of \(f(x)\) at the points given in the question: \(-2\), \(1\), \(4\), and \(1.5\).
- **At \(x = -2\)**: This is an integer point. The function is discontinuous here.
- **At \(x = 1\)**: This is also an integer point. The function is discontinuous here.
- **At \(x = 4\)**: This is another integer point. The function is discontinuous here.
- **At \(x = 1.5\)**: This is a non-integer point. The function is continuous here.

4. **Conclusion**:
The function \(f(x) = [x]\) is continuous at \(x = 1.5\) and discontinuous at the integer points \(-2\), \(1\), and \(4\).

### Final Answer:
The function \(f(x) = [x]\) is continuous at \(1.5\).