Consider the following function `f:RtoR` such that
`f(x)=x" if "xge0andf(x)=-x^(2)" if "xlt0`. Then, which one of the following is correct?

Options

A. f(x) is continuous at every `x""inR`
B. f(x) is continuous at x=0 only
C. f(x) is discontinous at x=0 only
D. f(x) is discontinuous at every `x""inR`

Explanation

To determine the continuity of the function \( f: \mathbb{R} \to \mathbb{R} \) defined as:

\[ f(x) = \begin{cases} x & \text{if } x \geq 0 \\ -x^2 & \text{if } x < 0 \end{cases} \]

we will check the continuity at \( x = 0 \) and analyze the behavior of the function for all \( x \in \mathbb{R} \).

### Step 1: Check the function at \( x = 0 \)

First, we need to find the value of the function at \( x = 0 \):

\[ f(0) = 0 \quad \text{(since \( 0 \geq 0 \))} \]

### Step 2: Calculate the left-hand limit as \( x \) approaches 0

Next, we calculate the left-hand limit:

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} -x^2 \]

As \( x \) approaches 0 from the left (negative side), we have:

\[ \lim_{x \to 0^-} -x^2 = -0^2 = 0 \]

### Step 3: Calculate the right-hand limit as \( x \) approaches 0

Now, we calculate the right-hand limit:

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

As \( x \) approaches 0 from the right (positive side), we have:

\[ \lim_{x \to 0^+} x = 0 \]

### Step 4: Compare the limits and the function value

Now we compare the left-hand limit, right-hand limit, and the function value at \( x = 0 \):

- Left-hand limit: \( \lim_{x \to 0^-} f(x) = 0 \)
- Right-hand limit: \( \lim_{x \to 0^+} f(x) = 0 \)
- Function value: \( f(0) = 0 \)

Since both the left-hand limit and right-hand limit are equal to the function value at \( x = 0 \), we conclude that:

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

### Conclusion

The function \( f(x) \) is continuous at \( x = 0 \).

### Step 5: Check continuity for \( x < 0 \) and \( x > 0 \)

For \( x < 0 \), \( f(x) = -x^2 \) is a polynomial function, which is continuous everywhere in its domain.

For \( x > 0 \), \( f(x) = x \) is also a polynomial function, which is continuous everywhere in its domain.

### Final Result

Since the function is continuous at \( x = 0 \) and continuous for all \( x < 0 \) and \( x > 0 \), we conclude that the function is continuous for all \( x \in \mathbb{R} \).

Thus, the correct option is:

**The function is continuous at every \( x \in \mathbb{R} \)**.

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