10. Given that functions $f(x), g(x)$ are periodic functions defined on $\mathbf{R}$, and they are monotonically increasing on the interval $[-1,1]$, which of the following statements is true:
A. $f(x)+g(x)$ is a periodic function
B. $f(x)+g(x)-|f(x)-g(x)|$ is monotonically increasing on $[-1,1]$
C. $f(g(x))$ is monotonically increasing on $[-1,1]$
D. $f(x)+g(x)+f(-x)+g(-x)$ is an even function
Among the correct propositions is $\qquad$
Let the functions $f(x), g(x)$ have periods $T_{1}, T_{2}$, respectively. Then the function $f(x)+g(x)$ is a periodic function if and only if $\frac{T_{1}}{T_{2}}$ is a rational number. Option $A$ is incorrect;
$f(x)+g(x)-|f(x)-g(x)|=\left\{\begin{array}{l}2 g(x), f(x) \geqslant g(x), \\ 2 f(x), f(x)<g(x)\end{array}=2 \min \{f(x), g(x)\}\right.$, combining the graph, we can see that option $B$ is correct;
According to the problem, the range of $g(x)$ for $x \in[-1,1]$ is not necessarily within the interval $[-1,1]$, so we cannot determine the monotonicity of the function $f(g(x))$, which means option $C$ is incorrect; $\square$
Option $D$ is obviously correct, so the answer is $B D$.