Let $f:\mathbb{R^2}\to \mathbb{R}$ such that for any pair $(x,y)\in\mathbb{R^2}$ the following holds: $$f(x+1,y)=f(x,y+1)=f(2x+y,x+y)=f(x,y)$$ Then if $f$ is continous, is it true that $f$ is constant?
And the second one, if $f$ is lebesgue integrable then it is constant almost everywhere?
I think both is true because this function is multi periodic but I don't know how to get it derictly from the equations above, can someone help?