Assume we define the locality of a theory in the following way:
Assume we have a theory of real scalars, so this theory is non local if the action has terms like $$\int d^dx\,\phi(x)V(x-y)\phi(y).$$ If the action does not have such interaction terms that are a product of the fields at different positions, the action is local.
So given this definition of locality assume a local free theory in continuum as $$S_{cnt}=\int dx^d\, (\partial_\mu\phi)^2.$$ In the lattice formulation, it becomes $$S_{lat}\propto\sum_{n\mu} \phi(na)[\phi(na+\mu)+\phi(na-\mu)-2\phi(na)].$$ So if the lattice spacing $a\to 0$, the lattice action would converge to the continuum action.
My question is:
Do the terms $\phi(na)\phi(na+\mu)$ break locality, with respect to the definition I have just provided?
