A function is analytic at a point c means the function have a convergent power series at point c and the function is infinitely differentiable at point c. For instance, if power series of the function converges to a constant. There is an example f(x)=|x|. It is not differentiable at x=0. It cannot even have MacLaurin series, which does not have a convergent power series at x=0.

It is useful, because it indicate us if a function-related question is able to solve by using the power series of it. As well, if a function is analytic at a point c, it is infinitely differentiable at point c; however, if a function is infinitely differentiable at point c, it may not be analytic at point c.