It will work for a while, but it can't work long term without other additions to the mix. Yeast need more than just sugar and water to live, basically, and they definitely need more to reproduce (a good source of nitrogen such as urea, a source of electrolytes like Na/K/Ca, and a bunch of other stuff in smaller quantities.) When you make wine or beer they can get most of this from the stuff you're fermenting, but even then you usually have to feed them to get them to reproduce well. Yeast also don't reproduce well without oxygen (the reason you aerate a wort or allow must to work for a day or two before starting the primary fermentation.) We don't aren't adding much oxygen at all when we add new sugar/water to the mix, though for the tiny amount of yeast we're using it might be sufficient.
I wind up replacing my yeast every 2nd or 3rd bottle, based on how it has been performing.