The set of all points where the function $f(x) = x + |x|$ is differentiable, is:

$(-\infty, 0) \cup (0, \infty)$

The correct answer is Option (3) → $(-\infty, 0) \cup (0, \infty)$ ##

$Lf'(0) = 0$ and $Rf'(0) = 2$; so, the function is not differentiable at $x = 0$.

For $x ≥ 0, f(x) = 2x$ (linear function) and when $x < 0$

$f(x) = 0$ (constant function).

Hence, $f(x)$ is differentiable when $x \in (-\infty, 0) \cup (0, \infty)$.