2. For any $x \in \mathbf{R}, f(x)=|\sin x|$, when $n \leqslant x$ $<n+1$ ( $n$ is an integer) then, $g(x)=x-n$. Among the 4 functions $f(x) 、 g(x) 、 f(x)+g(x) 、 f(x) g(x)$, the number of functions that are definitely periodic is ( ).
(A) 1
(B) 2
(C) 3
(D) 4
2.B.

It is easy to verify that the periods of $f(x)$ and $g(x)$ are $\pi$ and 1, respectively. Neither $f(x)+g(x)$ nor $f(x) g(x)$ are periodic functions.
In fact, if $f(x)+g(x)$ is a periodic function with period $T (T \neq 0)$, then for any $x \in \mathbf{R}$, we have
$$
f(x+T)+g(x+T)=f(x)+g(x) .
$$

In equation (1), let $x=0$, then
$$
f(T)+g(T)=f(0)+g(0)=0 .
$$

Also, $f(T) \geqslant 0, g(0) \geqslant 0$, so $f(T)=0, g(T)=0$.
From $f(T)=0$, we get $T=k \pi$ (where $k$ is a non-zero integer);
From $g(T)=0$, we get $T=m$ (where $m$ is a non-zero integer).
Therefore, $m=k \pi$. This is a contradiction.
Similarly, it can be proven that $f(x) g(x)$ is also not a periodic function.