When an function is analytic at c, it means that this function can be differentiable infinitely and it is converges to f(x) for x near c. This definition is useful because it can change a function into a infinite power series. This is more convenient for us to figure out many problems, such as whether this function converges or estimate decimal result.

A function differentiable infinitely does not mean it is analytic, such as problem.

However, an analytic function can be differentiable infinitely.