How the Smell Forms
You can identify sulfur by the smell of rotten eggs coming from your tap water as you turn on the faucet. If the smell stops after awhile, it doesn't mean that the sulfur doesn't exist anymore in your water supply.
This notorious smell of rotten eggs comes when sulfur dissolves into water, forming hydrogen sulfide gas and emitting it into the surrounding environment.
Bacteria That Form Sulfur
The sulfur in your water supply might come from bacteria that digests certain elements in water and creates sulfur as a byproduct of its digestion process. If you have a bacteria infestation in your house and your neighbor doesn't, the problem comes from within your home can be ruled out.
Call a professional to get your water purified with chlorine or ultraviolet purification to get rid of this problem.
Sulfuric Smells From Hot Water
If you have a smell coming from your faucet only when you use hot water, you need to get your water heater's sacrificial anode replaced. The anode prevents your heater's interior from corroding but may also begin to produce sulfur when it has aged significantly.
Not only do you get that ugly smell if you keep postponing the anode replacement, but you also put your water heater in danger of corrosion.
Water Softener Systems
Water softeners create problems when you turn them off after an extended period. If no water flows through the softener after a week or so, bacteria trapped in its reservoir might feed on the interior residue and form sulfuric deposits which cause that nasty smell in your supply.
You can tell if the water softener system causes the problem by testing your outside water spigots. If the spigot's water doesn't have any smell and the rest of your house's faucets have one, you must get your water softener sanitized as soon as possible.