# Extended transition function

($1)$2 | $3 ($4) | $5 ($6)
Given an DFA , we define the extended transition function inductively by , and .

The first part of this definition says that processing the empty string doesn't move the machine; the second part says that to process , you first process , and then take one more step using a from wherever gets you. Contrast the domain with that of , which processes only a single character. This distinction is important: since only processes characters, its domain is finite, so the description of the machine is finite; but (which is not part of the description of the machine) can process an infinite number of strings.