Why is the notion of analytic function so important?

Analytic functions have several nice properties, including but not limited to:

1. They are $C^\infty$ functions.
2. If, near $x_0$, we have $$f(x)=a_0+a_1(x-x_0)+a_2(x-x_0)^2+a_3(x-x_0)^3+\cdots,$$ then $$f'(x)=a_1+2a_2(x-x_0)+3a_3(x-x_0)^2+4a_4(x-x_0)^3+\cdots$$ and you can start all over again. That is, you can differentiate them as if they were polynomials.
3. The fact that you can express them locally as sums of power series allows you to compute fast approximate values of the function.
4. When the domain is connected, the whole function $f$ becomes determined by its behaviour in a very small region. For instance, if $f\colon\mathbb{R}\longrightarrow\mathbb R$ is analytic and you know the sequence $\left(f\left(\frac1n\right)\right)_{n\in\mathbb N}$, then this knowledge completely determines the whole function (the identity theorem).