The point(s), at which the function f given by `f(x) = {{:((x),xlt0),(-1,xge0):}` is continuous, is/are:

Options

A. `x in R`
B. `x =0`
C. `x in R-{0}`
D. `x = -1 and 1`

Answer

To determine the points at which the function \( f(x) \) is continuous, we need to analyze the function defined as:

\[ f(x) = \begin{cases} x & \text{if } x < 0 \\ -1 & \text{if } x \geq 0 \end{cases} \]

### Step 1: Understand Continuity
A function is continuous at a point \( a \) if:
\[ \lim_{x \to a} f(x) = f(a) \]
This means that the left-hand limit and the right-hand limit at that point must equal the function's value at that point.

### Step 2: Check Continuity at \( x = 0 \)
We need to check the continuity at the point \( x = 0 \) since the function has different definitions for \( x < 0 \) and \( x \geq 0 \).

#### Step 2.1: Calculate \( f(0) \)
From the definition of the function:
\[ f(0) = -1 \quad \text{(since \( 0 \geq 0 \))} \]

#### Step 2.2: Calculate the Right-Hand Limit
We calculate the limit as \( x \) approaches \( 0 \) from the right:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (-1) = -1 \]

#### Step 2.3: Calculate the Left-Hand Limit
Now we calculate the limit as \( x \) approaches \( 0 \) from the left:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x = 0 \]

### Step 3: Compare Limits and Function Value
Now we compare the limits and the function value:
- \( \lim_{x \to 0^+} f(x) = -1 \)
- \( \lim_{x \to 0^-} f(x) = 0 \)
- \( f(0) = -1 \)

Since the left-hand limit \( (0) \) does not equal the right-hand limit \( (-1) \), and the function value \( f(0) \) does not equal the left-hand limit, we conclude that:

\[ \text{The function is not continuous at } x = 0. \]

### Step 4: Determine Continuity Elsewhere
Since the function is defined as \( f(x) = x \) for \( x < 0 \) and \( f(x) = -1 \) for \( x \geq 0 \), we can see that:
- For \( x < 0 \), \( f(x) = x \) is a linear function and is continuous everywhere in that interval.
- For \( x > 0 \), \( f(x) = -1 \) is a constant function and is also continuous everywhere in that interval.

### Conclusion
Thus, the function \( f(x) \) is continuous for all \( x \) except at \( x = 0 \). Therefore, the points at which the function is continuous are:

\[ \text{The function is continuous for } x \in \mathbb{R} \setminus \{0\}. \]

### Final Answer
The correct option is: \( x \in \mathbb{R} - \{0\} \).

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