Liquid yeast saving

Is it possible to make your yeast last forever by adding it to a gallon of water with a teaspoon of sugar in it then when you need to use it taking a half of it off to pitch into your wort and keeping half of e yeast back from the original gallon then making it back up to a gallon with water and adding another teaspoon of sugar then repeating when it needs using...?

Probably in theory, but you don't want to feed yeast simple sugars. They, and your beer, will suffer for it.
Twice I have made an extra large starter, then pitched half of it and saved the other half. This is similar, you would just keep building and saving. I'm sure someone here does that.

Edit: Better yet, read the thread "can I just never buy yeast again"posted yesterday. Can't link on phone easy.

Doesn't sound like a good idea. Try harvesting/yeast washing. Maybe even freezing. Whatever you decide to do, you need to use starter wort made from malt extract or base malt, not simple sugar.