2. (8th "Hope Cup" Senior High School Competition) If functions $f_{1}(x)$ and $f_{2}(x)$ are both defined on $\mathbf{R}$ and are periodic functions with the smallest positive period $T$, for the function $y=f_{1}(x)+f_{2}(x)$, which of the following judgments is correct?
A. The smallest positive period is $T$
B. There is a smallest positive period $t$, and $t<T$
C. It is a periodic function, but may not have a smallest positive period
D. It may be a non-periodic function
2. C For any real number $x$, we have $f_{1}(x+T)=f_{1}(x), f_{2}(x+T)=f_{2}(x)$. Therefore, $f_{1}(x+T)+f_{2}(x+T)$ $=f_{1}(x)+f_{2}(x)$, so $y=f_{1}(x)+f_{2}(x)$ is a periodic function.

Take $f_{1}(x)=\sin x, f_{2}(x)=-\sin x$, then $y=f_{1}(x)+f_{2}(x)=0$ is a periodic function, but it has no smallest positive period (any non-zero number is its period), choose (.

Analysis of the periodicity of the sum of two functions is a very complex issue, and many problems remain unsolved. Generally, the discussion starts from the definition of a periodic function, combined with counterexamples.
The following are some conclusions about the periodicity of the sum of two functions:
（1）The sum of two periodic functions may be a periodic function or a non-periodic function. Let $f(x)=\sin x, g(x)=\sin \pi x$, $h(x)=1$, then the sum of two periodic functions $f(x)+h(x)=\sin x+1$ is a periodic function, while $f(x)+g(x)=\sin x+\sin \pi x$ is a non-periodic function.
（2）The sum of a periodic function and a non-periodic function may be a periodic function or a non-periodic function.
Let $f(x)=\sin x+\sin \pi x, g(x)=-\sin x \cdot h(x)=-2 \sin x$. Then $f(x)+g(x)=\sin \pi x$ is a periodic function, $f(x)$ $+h(x)=\sin \pi x-\sin x$ is a non-periodic function.
(3) If the functions $f_{1}(x)$ and $f_{2}(x)$ are both periodic functions, with the smallest positive periods being $T_{1}, T_{2}\left(T_{1} \neq T_{2}\right)$, when $T_{1} / T_{2}$ is a rational number, the function $y=f_{1}(x)+f_{2}(x)$ is a periodic function. The least common multiple of $T_{1}, T_{2}$ is one of its periods.