regarding DFA, δ(q, e) = δ( q). this is not a valid transition function because epsilon transitions are not defined for a DFA.
But we can consider it as a valid transition also, it means if we are not reading any input symbol then DFA remains on the same state.
Regarding extended transition function, in plcae of an input alphabet we provide string as an input for example
δ*(q0, ababa) = δ( q2). is an example of extended transition function