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?