Difference between domain of a function and domain ( a connected open set ) used in complex analysis . What is the difference between domain of a function and domain ( a connected open set ) used in complex analysis hugely? To be a domain of a function if the set should be always open and connected then my concept of a function is totally contradicted. So there should be a difference between these two domains. Please explain.
 A: Yes, these are slightly different uses of the word (just like how in any language the same word can have multiple meanings based on the context). The domain of a function is just a set. Apriori, no further restrictions are needed. The domain of a function can even be the following set of "symbols" $E=\{\ddot{\smile},\ddot{\frown}, \ddot{\smile}\ddot{\smile}, @, \#\}$. THe domain of a function does not have to be a subset of complex numbers or real numbers or anything.
The use of "domain" to mean a connected open subset of $\Bbb{C}$ is indeed a different meaning to above. I've also heard people use the term "region" to describe a connected open set. For example, people use this terminology when referring to things like "domain of integration/region of integration" etc.
Sometimes, of course, if you're considering a function $f:U\to\Bbb{C}$, where $U\subset \Bbb{C}$ is open and connected, then we can say "the domain of $f$ is a domain", where of course the two uses of the word "domain" in this same sentence have different meanings. Based on the context, you should be able to decipher which meaning is intended.
