Even or odd, periodic or not $k>0 \; ; \forall x\in\mathbb{R}, f(x+k)=f(k-x)\; and\; f(2k+x)=-f(2k-x) $

Question

I need to know if the function $f$ are odd or even, and I need to know if it periodic or not. $$k>0 \; ; \forall x\in\mathbb{R}, f(x+k)=f(k-x)\; and\; f(2k+x)=-f(2k-x) $$ I've tried to substitute x with different values this what I got : $$ x=0 \; ; f(2k)=-f(2k)$$ $$ x=k \; ; f(2k)=f(0)\; and\; f(3k)=-f(k) $$ $$f(3k)=f(2k+k)=f(k-2k)=f(-k)$$ Then, $$f(-k)=-f(k) $$ it's not enough to say $f$ is odd $(k>0)$ the function $f$ is defined on $\mathbb{R}$. Any help please.