Which of these functions is not continuous at $x = 0$?

Explanation

1. 01

   Just left of 0, $[x] = -1$, so $x + [x] \to -1$; at 0 and just right of it, $[x] = 0$, so $x + [x] \to 0$. A jump.

2. 02

   $x[x] \to 0 = f(0)$ from both sides, because the factor $x$ tends to 0; $x - \lvert x \rvert$ and $\sin \lvert x \rvert$ are built from continuous functions.