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On the Periodicity of Mathematical Functions
1. If f(x) is an even function and its image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 2a.

It is proved that f(x) is an even function, so f(x)=f(-x). F(x) is symmetrical about the straight line x=a, so f(x)=f(2a-x).

So f(x+2a)=f(-x+2a)=f(x).

Therefore, f(x) is a periodic function with a period of 2a.

2. If f(x) is a odd function whose image is symmetrical about the straight line x=a, then f(x) is a periodic function with a period of 4a.

It is proved that f(x) is odd function, so f(x) = -f(-x). F(x) is symmetrical about the straight line x=a, so f(x)=f(2a-x).

F(-x)=f(2a+x). Therefore, -f(-x)=- f(2a+x)= f(x). Furthermore, f(4a+x)= -f(2a+x)=f(x).

Therefore, f(x) is a periodic function with a period of 4a.

3. If f(x) is symmetric about points (a, 0) and (b, 0), then f(x) is a periodic function with a period of 2(b-a).

It is proved that f(x) is symmetric about point (a, 0), so f(x)=-f(2a-x).

F(x) is symmetrical about point (b, 0), so f(x)=-f(2b-x) =-f(2a-x).

So f(2a-x)=f(2b-x). Let 2a-x=y, then x = 2a-y. F(2a-x)=f(2b-x) becomes f(y)=f(2b-2a+y).

So f(x) is a periodic function with a period of 2(b-a).

4. If the image of f(x) is symmetrical (a ≠ b) about the straight line x = a and x = b, then f(x) is a periodic function with a period of 2(b-a).

It is proved that f(x) is symmetrical about the straight line x=a, so f(x)=f(2a-x).

F(x) is symmetrical about the straight line x=b, so f(x)=f(2b-x)=f(2b-x).

So f(2a-x)=f(2b-x). Let 2a-x=y, then x = 2a-y. F(2a-x)=f(2b-x) becomes f(y)=f(2b-2a+y).

So f(x) is a periodic function with a period of 2(b-a).