It's one of the definitions, and perhaps not the most fundamental. In the complex plane (and all real analytic functions can be analytically extended into the

It's one of the definitions, and perhaps not the most fundamental. In the complex plane (and all real analytic functions can be analytically extended into the complex plane), you can define the equivalent property as saying a (first!) derivative has the some value no matter how you chose to take the limit. So just by virtue of having a well defined first derivative in a particular region you get all the Maclaurin series magic.