The set of all points for which `f(x)=|x| |x-1| + 1/( " [ " x+1 " ] ") ` ([x] is the greatest integer function ) is continuous is

Options

A. `R`
B. `R~I`
C. `R~(I uu [-1,0))`
D. none of these

Answer

To determine the set of all points for which the function \( f(x) = |x| |x-1| + \frac{1}{[x+1]} \) is continuous, we will analyze the components of the function step by step.

### Step 1: Identify the components of the function
The function consists of two parts:
1. \( |x| |x-1| \)
2. \( \frac{1}{[x+1]} \) where \([x]\) is the greatest integer function.

### Step 2: Analyze the continuity of \( |x| |x-1| \)
The function \( |x| |x-1| \) is a product of continuous functions (absolute value functions), and thus it is continuous everywhere on the real line \( \mathbb{R} \).

### Step 3: Analyze the continuity of \( \frac{1}{[x+1]} \)
The term \( [x+1] \) is the greatest integer function, which is discontinuous at integer points. Specifically, \( [x+1] \) will be zero when \( x + 1 = 0 \) or \( x = -1 \). Thus, \( \frac{1}{[x+1]} \) is undefined at \( x = -1 \).

### Step 4: Determine when \( [x+1] \) is equal to zero
The function \( [x+1] \) is zero only at \( x = -1 \). For \( x \) in the interval \( (-1, 0) \), \( [x+1] = 0 \) and thus \( \frac{1}{[x+1]} \) is undefined in this interval. Therefore, the function \( f(x) \) is discontinuous for \( x \in (-1, 0) \).

### Step 5: Identify discontinuities due to the greatest integer function
The greatest integer function \( [x+1] \) is also discontinuous at all integer points. Thus, we need to check the integers:
- At \( x = 0 \), \( [0+1] = 1 \) (continuous)
- At \( x = 1 \), \( [1+1] = 2 \) (continuous)
- At \( x = -1 \), \( [0] = 0 \) (discontinuous)

### Step 6: Combine the results
From the analysis:
- The function \( f(x) \) is continuous everywhere except:
- At \( x = -1 \) (where \( \frac{1}{[x+1]} \) is undefined)
- In the interval \( (-1, 0) \) (where \( \frac{1}{[x+1]} \) is undefined)
- At all integer points where \( [x+1] \) is discontinuous.

Thus, the function is discontinuous at:
- \( x = -1 \)
- All integers \( \mathbb{Z} \)

### Final Answer
The set of all points for which \( f(x) \) is continuous is:
\[ \mathbb{R} \setminus \{ -1 \} \cup \mathbb{Z} \]