2. When processing data, we often round the obtained data. For any real number $x$, define a rounding function $f(x)$ accurate to the integer part, that is, if $n-0.5 \leqslant x < n+0.5 \ (n \in \mathbf{Z})$, then $f(x)=n$. For example, $f(3.49)=3, f(-1.2)=-1, f(-2.5)=-2$, etc. Given the following statements about the function $f(x)$:
(1) $f(x)$ is an odd function;
(2) $f(x)$ is an increasing function;
(3) For any $m \in \mathbf{Z}$, $f(x+m)=f(x)+m$;
(4) For any $x, y \in \mathbf{R}$, $f(x+y) \geqslant f(x)+f(y)$;
(5) For any $x, y \in \mathbf{R}$, $f(x+y) \leqslant f(x)+f(y)$.
Among them, the correct ones are $\qquad$ (fill in the sequence number).
(Jin Lei provided the question)
2. (3).
(3) is clearly correct. From $f(-0.5)=0, f(0.5)=1$, we know (1) is incorrect. From $f(0)=f(0.1)=0$, we know (2) is incorrect. Let $x=1.5, y=2.5$, we get $f(x+y)f(x)+f(y)$, so (5) is incorrect.