Serverless computing is inherently stateless, meaning that no state should be maintained on the host machine. This also means not sharing state between any parallel or sequential function executions. Any required state needs to be persisted to a database, a file server, or a cache.
In recent years, the stateless approach was made popular by the Twelve-Factor methodology, and many applications have already been refactored to use stateless web and logic tiers. The following quote is from the Twelve Factor App Methodology, factor 6: