[Math] the basic difference between differentiable, analytic and holomorphic function

Question

The function $f(z)$ is said to be analytic at $z_0$ if its derivative exists at each point $z$ in some neighborhood of $z_0$, and the function is said to be differentiable if its derivative exist at each point in its domain.
So whats the difference?

Answer 1

Holomorphic functions are complex functions, analytic functions are not necessarily complex.

Both, holomorphic and analytic functions, are infinitely continuous differentiable. But a differentiable functions is not necessarily infinitely differentiable, moreover: an infinitely differentiable function is not necessarily analytic or holomorphic.