Yes, they continue to function for at least several minutes. Firstly, what exactly is death? From a medical point of view, someone is dead when either their heart (circulation) and respiratory systems (breathing) stop in a way that can’t be reversed, or if their brain is so damaged that it can’t regain some very carefully defined functions. So, you can be dead in one of two different ways. However, the body is made of cells, and these definitions don’t tell us that all the cells are dead. Nerves are made up of axons, which are parts of cells called neurons. The death of these neurons is a different process from that of the death of the whole animal (including people!). So, someone can be brain dead, but kept alive on a ventilator, in which case nerve function can continue indefinitely, but this is very rare. If you heart stops, then the nerves can continue functioning (firing action potentials) for some minutes until the cells can no longer keep themselves alive. This is mainly due to them not being able to generate enough cellular energy to keep essential process in the cell working. Some nerves (such as the autonomic nerves I study) can survive for tens of minutes without a proper oxygen supply – this has been known for hundreds of years, where isolated organs have been taken from dead animals, electrically stimulated, and it has been shown that local functions (depending on nerves) remain for some time.