Unless the tenet is only going occupy the house a very brief time after the death of the owner who bequeathed you the house, I'd say you've inherited a rental and are just going to have to deal with it as such. Once the tenant leaves you take the house out of the rental pool and it becomes an investment house. You start "your" depreciation using the house's fair market value at the date of death.
I guess, too, you could immediately sell the rental to some other investor, subject to the lease.
