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my doubt is

what is the difference between transition function (delta) and extended transition function (delta cap ) in finite automata ?

both of them when started at a state q for a string w will lead to same sate p what is the difference

is transition functions gives the transition over an input alphabet , where as the extended transition symbol gives a transition for the string w using the transition function?

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    $\begingroup$ Please define the "extended transition function" -- it's not a concept I'm familiar with. And, I suggest that, if you write out the definition, the difference from the standard transition function should become obvious. $\endgroup$ – David Richerby Jun 22 '19 at 10:23
  • $\begingroup$ Please edit the question to add a reference to the textbook or article you are talking about. $\endgroup$ – John L. Jun 22 '19 at 12:15
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I read some references in order to answer your question ,

Transition function : takes as arguments a state and an input symbol and returns a state, denoted by δ . Extended transition function : Describes what happens when we start in any state and follow any sequence of inputs ,means is a function that takes a state q and a string w and returns a state p (the state that the automaton reaches when starting in state q and processing the sequence of inputs w). denoted by ˆδ . in simple term : transition function takes two parameters: a state and a symbol , while extended transition function takes a state and a string.

Example of transition function :

Example of extended transition function :

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