Let `f(x)={{:(x[(1)/(x)]+x[x],if, x ne0),(0,if,x=0):}` (where [x] denotes the greatest integer function). Then the correct statement is/are

Options

A. Limit exists for `x=-1`.
B. f(x) has a removable discontinuity at x = 1.
C. f(x) has a non removable discontinuity at x = 2.
D. f(x) is discontinuous at all positive integers.

Explanation

To solve the problem, we need to analyze the function given:

\[ f(x) = \begin{cases} x \left( \frac{1}{x} + [x] \right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \]

where \([x]\) denotes the greatest integer function.

### Step 1: Simplify the function for \( x \neq 0 \)

For \( x \neq 0 \):
\[ f(x) = x \left( \frac{1}{x} + [x] \right) = 1 + x[x] \]

### Step 2: Evaluate \( f(2) \)

To find \( f(2) \):
Since \( 2 \) is not equal to \( 0 \):
\[ f(2) = 1 + 2[2] = 1 + 2 \cdot 2 = 1 + 4 = 5 \]

### Step 3: Evaluate the left-hand limit as \( x \) approaches \( 2 \)

Calculate \( f(2^-) \):
As \( x \) approaches \( 2 \) from the left, \( [x] \) is \( 1 \):
\[ f(2^-) = \lim_{x \to 2^-} f(x) = 1 + 2[x] = 1 + 2 \cdot 1 = 1 + 2 = 3 \]

### Step 4: Evaluate the right-hand limit as \( x \) approaches \( 2 \)

Calculate \( f(2^+) \):
As \( x \) approaches \( 2 \) from the right, \( [x] \) is \( 2 \):
\[ f(2^+) = \lim_{x \to 2^+} f(x) = 1 + 2[x] = 1 + 2 \cdot 2 = 1 + 4 = 5 \]

### Step 5: Check continuity at \( x = 2 \)

Since \( f(2^-) = 3 \) and \( f(2^+) = 5 \), and \( f(2) = 5 \):
- The left-hand limit does not equal the right-hand limit, hence \( f(x) \) is discontinuous at \( x = 2 \).

### Step 6: Check for removable discontinuity at \( x = 1 \)

Evaluate \( f(1) \):
\[ f(1) = 1 + 1[1] = 1 + 1 \cdot 1 = 2 \]

Evaluate limits around \( x = 1 \):
- For \( x \to 1^- \), \( [x] = 0 \):
\[ f(1^-) = 1 + 1[0] = 1 + 0 = 1 \]
- For \( x \to 1^+ \), \( [x] = 1 \):
\[ f(1^+) = 1 + 1[1] = 1 + 1 = 2 \]

Since \( f(1^-) = 1 \) and \( f(1) = 2 \), we have a removable discontinuity at \( x = 1 \).

### Conclusion

The correct statements about the function \( f(x) \) are:
1. \( f(x) \) is discontinuous at all positive integers.
2. At \( x = 1 \), it has a removable discontinuity.