As mentioned above, earth and neutral are solidly connected at the source of supply in all public supply systems, and may in fact be the same conductor part or all of the way from there to your house.
On TN systems there will only be a fraction of an ohm of copper between your consumer unit and the connection. If it is TN-C-S that will be a very small fraction indeed because there will be a solid connection within your own service head. On a TT installation with the earth obtained via a rod / electrode, there might be a hundred ohms or more between earth and neutral at your end, but the connection is just as solid at the supply end.
The connection at source keeps the neutral near earth potential and is what makes it a neutral (as far as single-phase installations are concerned.) The other wire, the one that is not connected to earth, then becomes the line.
If the continuity disappears when you turn off the main switch, there is no indication that your installation is faulty, this is what you would expect. However, despite being near earth potential, neutral is considered a live conductor (it is part of the energy supply circuit) and you should not go poking around at it with the continuity tester with the main switch on. In any case, whatever reading you get is somewhat arbitrary because there is usually some AC voltage present (due to voltage drop along conductors) and most continuity and resistance tests are not supposed to be carried out with stray voltages present.