Consider the following function `f:Rrarr R` such that f(x) = x, if `xge0` and `f(x)=-x^(2)`, if `xlt0` then, which one of the following is correct?

Options

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

Answer

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 the point \( x = 0 \) since the function has different definitions on either side of this point.

### Step 1: Find the left-hand limit as \( x \) approaches 0.

For \( x < 0 \), the function is defined as \( f(x) = -x^2 \). We need to calculate:

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

Substituting \( x = 0 \):

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

### Step 2: Find the right-hand limit as \( x \) approaches 0.

For \( x \geq 0 \), the function is defined as \( f(x) = x \). We need to calculate:

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

Substituting \( x = 0 \):

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

### Step 3: Find the value of the function at \( x = 0 \).

We need to evaluate \( f(0) \):

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

### Step 4: Check if the left-hand limit, right-hand limit, and the value of the function at \( x = 0 \) are equal.

We have:

- Left-hand limit: \( \lim_{x \to 0^-} f(x) = 0 \)
- Right-hand limit: \( \lim_{x \to 0^+} f(x) = 0 \)
- Value of the function at \( x = 0 \): \( f(0) = 0 \)

Since all three values are equal, we conclude that:

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

### Conclusion

The function \( f(x) \) is continuous at \( x = 0 \). Since the function is continuous for all \( x < 0 \) (as it is a polynomial) and for all \( x > 0 \) (as it is also a linear function), we can conclude that the function is continuous everywhere on \( \mathbb{R} \).

Thus, the correct answer is:

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

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