If a person has no brain activity in their brain, then that person may be considered brain dead. Brain dead is a legal indicator of death in some states. without brain activity, a person most likely will be unable to keep their body functioning without some form of life support, their major organs would start to fail.
There has been cases of a babies born with just a brain stem. They need to take medications to survive, such as seizure meds. Nicolas Coke I believe lived to be 2 years old but I am unsure if he is still alive.