If `f(x)={:{([x]+[-x] , xne 0), ( lambda , x =0):}` where [.] denotes the greatest function , then f(x) is continuous at ` x =0 , " for" lambda `

Options

A. `-1`
B. 0
C. 1
D. No value is possible

Answer

To solve the problem, we need to determine the value of \( \lambda \) such that the function \( f(x) \) is continuous at \( x = 0 \). The function is defined as follows:

\[ f(x) = \begin{cases} [x] + [-x] & \text{if } x \neq 0 \\ \lambda & \text{if } x = 0 \end{cases} \]

where \([x]\) denotes the greatest integer function (also known as the floor function).

### Step 1: Evaluate the limit as \( x \) approaches 0 from the right (\( x \to 0^+ \))

For \( x \) approaching 0 from the right, \( x \) is a small positive number. Therefore, we have:

\[ [x] = 0 \quad \text{(since } x \text{ is between 0 and 1)} \]
\[ [-x] = -1 \quad \text{(since } -x \text{ is between -1 and 0)} \]

Thus,

\[ f(x) = [x] + [-x] = 0 + (-1) = -1 \quad \text{for } x \to 0^+ \]

### Step 2: Evaluate the limit as \( x \) approaches 0 from the left (\( x \to 0^- \))

For \( x \) approaching 0 from the left, \( x \) is a small negative number. Therefore, we have:

\[ [x] = -1 \quad \text{(since } x \text{ is between -1 and 0)} \]
\[ [-x] = 0 \quad \text{(since } -x \text{ is between 0 and 1)} \]

Thus,

\[ f(x) = [x] + [-x] = -1 + 0 = -1 \quad \text{for } x \to 0^- \]

### Step 3: Evaluate \( f(0) \)

From the definition of the function, we have:

\[ f(0) = \lambda \]

### Step 4: Set the limits equal to ensure continuity

For \( f(x) \) to be continuous at \( x = 0 \), we need:

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^-} f(x) = f(0) \]

From our previous evaluations, we found:

\[ \lim_{x \to 0^+} f(x) = -1 \]
\[ \lim_{x \to 0^-} f(x) = -1 \]
\[ f(0) = \lambda \]

Setting these equal gives:

\[ -1 = \lambda \]

### Conclusion

Thus, the value of \( \lambda \) that makes \( f(x) \) continuous at \( x = 0 \) is:

\[ \lambda = -1 \]