At how many points is this function not differentiable?

Explanation

1. 01

   $x\lvert x \rvert$ is $-x^{2}$ for $x < 0$ and $x^{2}$ for $x \ge 0$: slope 0 on both sides of 0, so no corner there.

2. 02

   $\lvert x - 1 \rvert$ has a corner at 1, and the smooth $x\lvert x \rvert$ cannot remove it: 1 point.