Discuss the differentiability of `f(x)=x|x|` at `x=0`

Explanation

To discuss the differentiability of the function \( f(x) = x |x| \) at \( x = 0 \), we will follow these steps:

### Step 1: Simplify the Function
The function \( f(x) = x |x| \) can be expressed differently based on the value of \( x \):
- For \( x < 0 \): \( |x| = -x \), so \( f(x) = x(-x) = -x^2 \).
- For \( x \geq 0 \): \( |x| = x \), so \( f(x) = x(x) = x^2 \).

Thus, we can rewrite the function as:
...