What is wrong about defining a connection to a DB for each function (each URL) in a web application?

I'd like to make the question generic and I dont specify any framework name. Actually I'm trying to get rid of the framework or to use a minimal framework, and I'd like to know what is wrong about defining a new connection to the database in each function ? I mainly want to know about performance implications, not the fact that its too much repetition. And I need to find out how to initialise the connection to the database when the application starts.