Assume we define the locality of a theory as following:

Asumme we have a theory of real scalars, so this theory is non local if the action has terms like

$$\int d^dx\phi(x)V(x-y)\phi(y)$$,

and if the action does not have such interaction terms that is 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,

$$S_{cnt}=\int dx^d (\partial_\mu\phi)^2$$

and 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 lattice spacing $a\to 0$ lattice action would converge to continuum action.

My question is: Are the terms $\phi(na)\phi(na+\mu)$ breaks the locality, with respect to the definition I have just provided?