It's possible to cut out sugar, yes. But also keep in mind sugar is literally everywhere and the cheapest easiest food substance to access. It's on every menu, it's at the check out counter of every store, it's offered for free by coworkers bringing in treats or birthday cakes, offered in bowls at the exit from some restaurants or other businesses, it's thrown out at parades, given away on Halloween, etc. It's possible to avoid eating sugar, but avoiding temptation is basically impossible if you have a habit of leaving your home to work or shop. If you are addicted to sugar, it's a constant struggle.

Do you mean added sugar? Because cutting out fruit and veg from your diet is pretty unhealthy.