A function to be analytic at c means that the function is differetiable for many times and the function has a convergent power series at the point c. For example, the absolute value function is not analytic because we cannot differentiate the function at x=0.

The analytical function is useful because it has a convergent power series at some point. Therefore, it is useful when we are solving the questions. Also, we can distinguish the analytical function by taking a derivative of the function at some point.