When a sole proprietor passes away, the business ends--the sole proprietor *was* the business, and there was no separate legal entity (as there would have been with a corporation, for example) to survive him. This does not mean that someone else can't buy the business's assets, including its name and customer/client list, from any heirs of the sole proprietor or the sole proprietor's estate and carry on--but he/she would be starting a new business that happens to use the assets (and even name) of the old, and not continuing the old business, which from a legal point of view, passed with the sole proprietor.