`f(x)={{:(,[x]+[-x], x ne 0),(,lambda,x=0):}`
Find the value of `lambda`, f is continuous at x=0?

Answer

To find the value of \( \lambda \) such that the function \( f(x) \) is continuous at \( x = 0 \), we need to ensure that the left-hand limit and right-hand limit at \( x = 0 \) are equal to \( f(0) \).

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 less than or equal to \( x \).

### Step 1: Calculate \( f(0) \)
Since \( f(0) = \lambda \), we will need to find the limits as \( x \) approaches 0.

### Step 2: Find the Right-Hand Limit as \( x \to 0^+ \)
We calculate the right-hand limit:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} ([x] + [-x]) \]

For \( x \) approaching 0 from the positive side, \( [x] = 0 \) (since \( x \) is between 0 and 1) and \( [-x] = -1 \) (since \( -x \) is between -1 and 0). Therefore:
\[ \lim_{x \to 0^+} f(x) = [0] + [-0] = 0 + (-1) = -1 \]

### Step 3: Find the Left-Hand Limit as \( x \to 0^- \)
Now we calculate the left-hand limit:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} ([x] + [-x]) \]

For \( x \) approaching 0 from the negative side, \( [x] = -1 \) (since \( x \) is between -1 and 0) and \( [-x] = 0 \) (since \( -x \) is between 0 and 1). Therefore:
\[ \lim_{x \to 0^-} f(x) = [-1] + [0] = -1 + 0 = -1 \]

### Step 4: Set the Limits Equal to \( f(0) \)
For \( f(x) \) to be continuous at \( x = 0 \), we need:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^-} f(x) = f(0) \]
This gives us:
\[ -1 = -1 = \lambda \]

### Conclusion
Thus, the value of \( \lambda \) such that \( f \) is continuous at \( x = 0 \) is:
\[ \lambda = -1 \]