1. The domain of the function $f(x)$ is $\mathbf{R}$, for any $x_{1}, x_{2} \in \mathbf{R}$, there is $f\left(x_{1}+x_{2}\right)=f\left(x_{1}\right)+f\left(x_{2}\right)$, when $x \geqslant 0$, $f(x)$ is an increasing function. Then among the following statements, the correct one is
A. $f(x)$ is an odd function, and it is increasing on $\mathbf{R}$
B. $f(x)$ is an even function, and it is increasing on $\mathbf{R}$
C. $f(x)$ is an odd function, but its monotonicity on $\mathbf{R}$ is uncertain
D. $f(x)$ is an even function, but its monotonicity on $\mathbf{R}$ is uncertain
1. Let $x_{2}=0, f(0)=0$; let $x_{1}=x, x_{2}=-x, f(-x)=-f(x)$, so $f(x)$ is an odd function. And when $x \geqslant 0$, $f(x)$ is an increasing function. Therefore, $f(x)$ is also an increasing function on $\mathbf{R}$.