The function` f(x)=x-[x]`,where`[ ]`denotes the greatest integer function, is :

Options

A. continuous everywhere
B. continuous at integer points only
C. continuous at non-integer points only
D. nowhere continuous

Answer

To determine the continuity of the function \( f(x) = x - [x] \), where \([x]\) denotes the greatest integer function, we will analyze the function step by step.

### Step 1: Understanding the Function
The function \( f(x) = x - [x] \) gives the fractional part of \( x \). This means:
- For any integer \( n \), \( f(n) = n - n = 0 \).
- For any non-integer \( x \) in the interval \( [n, n+1) \), where \( n \) is an integer, \( f(x) = x - n \), which is a linear function ranging from \( 0 \) to \( 1 \) as \( x \) increases from \( n \) to \( n+1 \).

### Step 2: Check Continuity at Integer Points
To check continuity at integer points, we need to evaluate the left-hand limit (LHL), right-hand limit (RHL), and the value of the function at that point.

1. **At \( x = n \) (where \( n \) is an integer)**:
- \( f(n) = 0 \) (since \( f(n) = n - n = 0 \)).
- **Left-hand limit** as \( x \) approaches \( n \) from the left (\( n^- \)):
\[ f(n^-) = n - 1 \quad \text{(since \( n-1 < n < n \))} \]
- **Right-hand limit** as \( x \) approaches \( n \) from the right (\( n^+ \)):
\[ f(n^+) = n - n = 0 \]

Therefore,
\[ \text{LHL} = 1 \quad \text{and} \quad \text{RHL} = 0 \]
Since LHL ≠ RHL, \( f(x) \) is discontinuous at integer points.

### Step 3: Check Continuity at Non-Integer Points
For non-integer points, say \( x \) in the interval \( (n, n+1) \):
- Here, \( f(x) = x - n \), which is a linear function and is continuous in this interval.

### Conclusion
- The function \( f(x) \) is discontinuous at all integer points.
- The function is continuous at all non-integer points.

Thus, the correct answer is that the function is continuous at non-integer points only.

### Final Answer
The function \( f(x) = x - [x] \) is continuous at non-integer points only.