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

Options

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

Explanation

To solve the problem, we need to analyze the function \( f(x) = [|x|] - |[x]| \), where \( [x] \) denotes the greatest integer function (also known as the floor function).

### Step-by-Step Solution:

1. **Understanding the Components**:
- The greatest integer function \( [x] \) returns the largest integer less than or equal to \( x \).
- The absolute value function \( |x| \) returns the non-negative value of \( x \).

2. **Analyzing \( |x| \)**:
- For any real number \( x \):
- If \( x \geq 0 \), then \( |x| = x \).
- If \( x < 0 \), then \( |x| = -x \).

3. **Analyzing \( [|x|] \)**:
- Since \( |x| \) is always non-negative, \( [|x|] \) will be the greatest integer less than or equal to \( |x| \).

4. **Analyzing \( |[x]| \)**:
- The value of \( [x] \) can be either positive, negative, or zero:
- If \( x \) is an integer, \( [x] = x \).
- If \( x \) is not an integer, \( [x] \) is the largest integer less than \( x \).
- Thus, \( |[x]| \) will be:
- \( [x] \) if \( [x] \geq 0 \) (i.e., \( x \geq 0 \))
- \(-[x]\) if \( [x] < 0 \) (i.e., \( x < 0 \))

5. **Finding \( f(x) \)**:
- For \( x \geq 0 \):
\[ f(x) = [|x|] - |[x]| = [x] - [x] = 0 \]
- For \( x < 0 \):
\[ f(x) = [|x|] - |[x]| = [-x] - (-[x]) = [-x] + [x] \]
Here, \( [-x] = -[x] - 1 \) (because \( -x \) is positive and \( [x] \) is negative), thus:
\[ f(x) = (-[x] - 1) + [x] = -1 \]

6. **Conclusion**:
- Therefore, the function can be summarized as:
\[ f(x) = \begin{cases} 0 & \text{if } x \geq 0 \\ -1 & \text{if } x < 0 \end{cases} \]

7. **Discontinuity**:
- The function \( f(x) \) is discontinuous at \( x = 0 \) because:
- As \( x \) approaches \( 0 \) from the left, \( f(x) \to -1 \).
- As \( x \) approaches \( 0 \) from the right, \( f(x) \to 0 \).
- Therefore, \( f(x) \) is continuous for all \( x \) except at \( x = 0 \).

### Final Result:
The function \( f(x) \) is continuous for all non-integer points and is discontinuous at integer points, particularly at \( x = 0 \).