Proving that a function is periodic

Question

I need help proving the following:

Let $f(x)$ be an even function and let $A$ be an arbitrary real number . If the function $g(x) = f(A - x) $ is odd then $f(x)$ is periodic.

Answer 1

Using $g(-x)=-g(x)$ we get that $$ f(A+x)=-f(A-x) $$

using that $f$ is even we get that $$ -f(A-x)=-f(x-A) $$

and so we have $$ f(x+A)=-f(x-A) $$

this is true for all $x$. Putting $x-A$ in the above we get that $$ f(x)=-f(x-2A)=f(x-4A) $$

hence $f$ is periodic