Explicit example of a discontinuous function between convergence spaces with the preimage of every open set open

Question

The definition of continuity for convergence spaces is in terms of filters, but it’s still true that the preimage of any open set is open. Since this is no longer taken as the def of continuity, I wanted to see if there’s an explicit example of a discontinuous function between convergence spaces that still satisfies “preimage of every open set is open”. Of course, these spaces would have to be non-topological.