If my mother recently passed away and had a morgage on her home, can I take up payments and keep it?

You can keep the house, but only if you follow procedures to get the house and mortgage in your name and it's an authorized settlement of the estate.

First, you need to find out if there is a will.  If the will awards you the home, subject to the mortgage, the you would need to probate the estate and work with the mortgage company to have the paperwork changed over into your name.  This may require a new note in your name instead of the estate's name.  If there is a will and it leaves the house to someone else, then, no, you would not be allowed to keep the house unless the other recipient was willing to give it to you.

If there is no will and no other heirs, you would still want to have the estate probated and have the probate court officially gift you the house.  And again, you would need to work with the mortgage company to get the information changed over to your name.  If there are other heirs, then you would need to reach some sort of agreement with the others about you keeping the home.

You don't want to just start making payments without going through the probate procedures, especially if there are other potential heirs.  If you make payments on the mortgage or improvements on the house, the heirs could come in later and demand a sell of the house and short-change you from some or all of the effort you placed into the house.

You want to make sure that the title and ownership pass clearly to go.  Consult with a probate attorney to make sure that the procedures are followed to insure a proper transfer.