Real functions may or may not have derivatives. Furthermore, existence of a derivative in one point does not assure its existence anywhere else. A real function may have one or two or, for that matter, any finite number of derivatives. Functions analytic in the whole plane are called entire . Polynomials are entire functions.
Analytic functions have derivatives of any order which themselves turn out to be analytic functions.
For analytic functions, this property is expressed by the Cauchy Integral Formula . Functions which are not entire have singularities in the finite plane. Entire functions have a singularity at infinity. The only ones that do not are constant . Polynomials have a pole at infinity.