Translation-invariant systems and binary functions of spatial variables

[hokhani]

TL;DR Summary

the correctness of the relation f(r,r')=f(r-r') in translation-invariant systems

My question is about a statement in the book, Many-Body Quantum Theory in Condensed Matter Physics: An Introduction, by Henrik Bruus, Karsten Flensberg in appendix A:
In a translation invariant system, any physical observable $f(\vec r,\vec r')$ of two spatial variables $\vec r$ and $\vec r'$ can only depend on the difference between the coordinates and not on the absolute position of any of them, $$f(\vec r,\vec r')=f(\vec r - \vec r').$$ However, I think in one unit cell if we shift identically both $\vec r$ and $\vec r'$ then this statement does not seem necessarily correct!

 

[BvU]

I think in one unit cell if we shift identically both $\vec r$ and $\vec r'$ then this statement does not seem necessarily correct!

What do you mean with that ? The statement doesn't mention unit cells.
And what is 'shift identically' ?

$\$

 

[anuttarasammyak]

TL;DR Summary: the correctness of the relation f(r,r')=f(r-r') in translation-invariant systems

coordinates and not on the absolute position of any of them, f(r→,r→′)=f(r→−r→′). However, I think in one unit cell if we shift identically both r→ and r→′ then this statement does not seem

For any position vector R
$$f(\mathbf{r}, \mathbf{r'})=f(\mathbf{r}+\mathbf{R}, \mathbf{r'}+\mathbf{R})$$
choosing R=-r or R=-r'
$$=f(\mathbf{r}-\mathbf{r'},0))=f(0, \mathbf{r'}-\mathbf{r}):=F(\mathbf{r}-\mathbf{r'})$$