Proving Periodicity of a Function

Question

How to prove that a function $f(x)$ symmetric with respect to the lines $x=a$ and $x=b$, where $a\neq b$) is periodic?

Any help or hint is welcome. Thanks in advance.

Answer 1

Note that $$f(x)=f(a-x),\tag{1}$$ and $$f(t)=f(b-t).\tag{2}$$ By putting $t = b-a+x$ in $(2)$, we get $$f(x+(b-a))=f(a-x)=f(x).\tag{3}$$