The function $f(x) = x|x|, x \in \mathbb{R}$ is differentiable:

only at $x = 0$

The correct answer is Option (1) → only at $x = 0$ ##

Given $f(x) = x|x|$, we rewrite it as:

$f(x) = \begin{cases} x^2, & x \ge 0 \\ -x^2, & x < 0 \end{cases}$

Check Differentiability at $x = 0$:

Left derivative: $f'(x) = -2x ⇒\lim\limits_{x \to 0^-} f'(x) = 0$.

Right derivative: $f'(x) = 2x ⇒\lim\limits_{x \to 0^+} f'(x) = 0$.

Since LHD = RHD, $f(x)$ is differentiable at $x = 0$.