The function f(x) = x |x| at x= 0 is

Options

A. continuous but not differentiable
B. differentiable but not continuous
C. continuous and differentiable
D. neither continuous nor differentiable

Answer

To determine whether the function \( f(x) = x |x| \) is continuous and differentiable at \( x = 0 \), we will follow these steps:

### Step 1: Define the function based on the value of \( x \)

The function can be expressed as:
- For \( x \geq 0 \):
\[ f(x) = x \cdot |x| = x^2 \]
- For \( x < 0 \):
\[ f(x) = x \cdot |x| = x \cdot (-x) = -x^2 \]

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

To check if \( f(x) \) is continuous at \( x = 0 \), we need to verify the following condition:
\[ \lim_{x \to 0} f(x) = f(0) \]

First, we calculate \( f(0) \):
\[ f(0) = 0 \cdot |0| = 0 \]

Next, we find the left-hand limit (LHL) and right-hand limit (RHL) as \( x \) approaches \( 0 \):

**Left-hand limit (LHL)**:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x^2) = 0 \]

**Right-hand limit (RHL)**:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x^2) = 0 \]

Since both limits equal \( f(0) \):
\[ \lim_{x \to 0} f(x) = 0 = f(0) \]
Thus, \( f(x) \) is continuous at \( x = 0 \).

### Step 3: Check differentiability at \( x = 0 \)

To check if \( f(x) \) is differentiable at \( x = 0 \), we need to find the left-hand derivative (LHD) and right-hand derivative (RHD) at \( x = 0 \).

**Left-hand derivative (LHD)**:
\[ \text{LHD} = \lim_{x \to 0^-} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0^-} \frac{-x^2 - 0}{x} = \lim_{x \to 0^-} -x = 0 \]

**Right-hand derivative (RHD)**:
\[ \text{RHD} = \lim_{x \to 0^+} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0^+} \frac{x^2 - 0}{x} = \lim_{x \to 0^+} x = 0 \]

Since both derivatives are equal:
\[ \text{LHD} = \text{RHD} = 0 \]
Thus, \( f(x) \) is differentiable at \( x = 0 \).

### Conclusion

The function \( f(x) = x |x| \) is both continuous and differentiable at \( x = 0 \).

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