A function f `RrarrR` is defined as `f(x)=x^(2)` for `xge0` and `f(x)=-x` for `xlt0` consider the following statements in respect to the above function
1. the function is continuous at x = 0.
2. the function is differentiable at x = 0
which of the above function is/are continuous?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To solve the problem, we need to analyze the function \( f(x) \) defined as follows:

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

We need to determine the continuity and differentiability of this function at \( x = 0 \).

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

A function is continuous at a point if the following three conditions are satisfied:
1. \( f(0) \) is defined.
2. The limit of \( f(x) \) as \( x \) approaches 0 exists.
3. The limit of \( f(x) \) as \( x \) approaches 0 equals \( f(0) \).

**Calculating \( f(0) \):**
\[ f(0) = 0^2 = 0 \]

**Calculating the Left-Hand Limit (LHL) as \( x \) approaches 0:**
\[ \text{LHL} = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x) = -0 = 0 \]

**Calculating the Right-Hand Limit (RHL) as \( x \) approaches 0:**
\[ \text{RHL} = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x^2) = 0 \]

Since both the left-hand limit and right-hand limit equal \( f(0) \):
\[ \text{LHL} = \text{RHL} = f(0) = 0 \]
Thus, \( f(x) \) is continuous at \( x = 0 \).

### Step 2: Check Differentiability at \( x = 0 \)

A function is differentiable at a point if the left-hand derivative (LHD) and right-hand derivative (RHD) at that point are equal.

**Calculating the Left-Hand Derivative (LHD) at \( x = 0 \):**
\[ \text{LHD} = \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^-} \frac{-h - 0}{h} = \lim_{h \to 0^-} -1 = -1 \]

**Calculating the Right-Hand Derivative (RHD) at \( x = 0 \):**
\[ \text{RHD} = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^+} \frac{h^2 - 0}{h} = \lim_{h \to 0^+} h = 0 \]

Since the left-hand derivative and right-hand derivative are not equal:
\[ \text{LHD} = -1 \quad \text{and} \quad \text{RHD} = 0 \]
Thus, \( f(x) \) is not differentiable at \( x = 0 \).

### Conclusion

1. The function is continuous at \( x = 0 \) (True).
2. The function is not differentiable at \( x = 0 \) (False).

Therefore, the correct statement is:
- **1 only** (the function is continuous at \( x = 0 \)).