Whew! This is tricky and it demonstrates the importance of having your financial affairs in order. First, it depends on whether you are on the TITLE, as well as the mortgage. If the house was titled in “joint name with rights of survivorship? you would have no problem. It would be fully yours, and you would still be on the hook for the mortgage payments since you were on the note, as well. OR, if the house were titled in the name of her Revocable Living Trust, and you were named a successor trustee (and inheritor, presuming no others listed) then you would still be ok — though still liable for payments on the note. And yes, you would notify the bank and present a death certificate so the title could be changed easily into your name — or the name of your OWN revocable living trust.
BUT, if she died without a will (and absent the two possibilities suggested above), then you need an estate attorney to help you though this process. The estate would have to go through probate, which requires a lawyer. In the meantime, you would keep paying the mortgage(assuming you can afford it, and assuming the house is worth more than the amount of the mortgage). The bank won’t make any moves to “kick you out” as long as it is being paid promptly. But again, consult an attorney to represent you on this issue. Check your local bar association directory for an attorney who specializes in estate planning.