I was asked to do more deeper question related to question 2 in the assignment.

For instance f(x)=sin(x) and g(x)=e^x, f(x) is analytic at c, and g(x) is not.

A function analytic at c means this function convergent power series for x=c. And only if Taylor series for particular function converge to this function for every single c at x=c.

Analytical functions are supposed to be infinite differentiable. It does not mean all the infinitely function is analytic, such as y=e^(-c)cos(cx).

The reason for its usefulness is that if we know that a function is analytic when x=c, we can get power series at x=c in domain.