Consider `f(x)={{:([x]+[-x],xne2),(lambda,x=2):}` where `[.]` denotes the greatest integer function. If `f(x)` is continuous at `x=2` then the value of `lambda` is equal to

Options

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

Answer

To solve the problem, we need to determine the value of \(\lambda\) such that the function \(f(x)\) is continuous at \(x = 2\). The function is defined as follows:

\[ f(x) = \begin{cases} [x] + [-x] & \text{if } x \neq 2 \\ \lambda & \text{if } x = 2 \end{cases} \]

where \([x]\) denotes the greatest integer less than or equal to \(x\).

### Step 1: Find the limit as \(x\) approaches 2 from the right (\(2^+\))

To find \( \lim_{x \to 2^+} f(x) \):
- For \(x\) just greater than 2 (e.g., \(x = 2.01\)):
- \([x] = [2.01] = 2\)
- \([-x] = [-2.01] = -3\) (since \(-2.01\) is between \(-3\) and \(-2\))

Thus,
\[ f(x) = [x] + [-x] = 2 + (-3) = -1 \]
So,
\[ \lim_{x \to 2^+} f(x) = -1 \]

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

To find \( \lim_{x \to 2^-} f(x) \):
- For \(x\) just less than 2 (e.g., \(x = 1.99\)):
- \([x] = [1.99] = 1\)
- \([-x] = [-1.99] = -2\) (since \(-1.99\) is between \(-2\) and \(-1\))

Thus,
\[ f(x) = [x] + [-x] = 1 + (-2) = -1 \]
So,
\[ \lim_{x \to 2^-} f(x) = -1 \]

### Step 3: Set the limits equal to \(f(2)\)

For the function \(f(x)\) to be continuous at \(x = 2\), we need:
\[ \lim_{x \to 2^+} f(x) = \lim_{x \to 2^-} f(x) = f(2) \]
From our calculations:
\[ -1 = -1 = \lambda \]

### Conclusion

Thus, the value of \(\lambda\) that makes \(f(x)\) continuous at \(x = 2\) is:
\[ \lambda = -1 \]