Two questions about the proof of periodic function 1. It is known that f (x) is an odd function, and the image of F (x) is symmetric with respect to the line x = 2. It is proved that f (x) is a periodic function 2. Let f (x) be an even function defined on R whose image is symmetric with respect to the line x = 1. For any x 1, x 2 belonging to [0,0.5], f (x 1 + x 2) = f (x 1) * f (x 2) is proved to be a periodic function

Two questions about the proof of periodic function 1. It is known that f (x) is an odd function, and the image of F (x) is symmetric with respect to the line x = 2. It is proved that f (x) is a periodic function 2. Let f (x) be an even function defined on R whose image is symmetric with respect to the line x = 1. For any x 1, x 2 belonging to [0,0.5], f (x 1 + x 2) = f (x 1) * f (x 2) is proved to be a periodic function

f(x)=-f(-x)
Since x = 2 is symmetric
f(2+x)=f(2-x)
It can also be said that
f(x+4)=f(-x)=-f(x)
f(x+8)=-f(x+4)=f(x)
So it's a function with period 8
2.f(x+2)=f(-x)=f(x)
So it's a function with period 2. It seems that the following conditions are useless, except that we can find f (0)