Why is a sum of an even and a periodic functions and $f(x)=x$ definitely not even?

Question

$f(x)$ is defined on $\mathbb{R}$, not constant and even. $g(x)$ is defined on $\mathbb{R}$, not constant and periodic.

Why is $f(x) + g(x) + x$ then not even?

It is known that a sum of an even and an odd functions ($f(x) + x$) is neither odd nor even.

But why does adding a periodic function definitely not turn the result into an even function?

Answer 1

Suppose $f(x)+g(x)+x$ was even. Since $f(x)$ is even, this implies $g(x)+x$ is even, i.e., for all $x$, $$g(x)+x=g(-x)-x$$ or equivalently $$g(x)-g(-x)=-2x$$ Now let $p>0$ be the period of $g$. Then $g(-p/2)=g(p/2)$, so setting $x=p/2$ above yields $$p=0$$ a contradiction (note that we did not use the fact that $f$ and $g$ are not constant).

Answer 2

You have it wrong: the sum of an even, an odd, and a periodic function can be even. You can easily see this by choosing $\sin x$ as your odd function and $-\sin x$ as your periodic function.