I put some mint cuttings in a plastic cup, along with tap water and a few pebbles. It's been a few months now, and the mint is still alive and kicking. Rather than getting some soil and a pot for the mint, would it be possible for the mint to survive indefinitely given adequate water and sunlight? Or would the mint eventually die from lack of nutrition?
Mint will last a long time in water. Give it a bit of balanced fertilizer and keep changing the water. Nothing lasts indefinitely. You could transplant your mint into soil, sterilized potting soil, not out in the garden yet. I'd not use larger than a 4" pot, allow your starts to root fully, you should be able to see roots coming through the bottom then transplant into your garden.