1. Analytic functions.
Definition  A function f is analytic at zo if the following conditions are fulfilled:
1. f is derivable at z0 ;
2. There exists a neighborhood V of zo such that f is derivable at every point of V .
Definition A function which is analytic on the whole of C is called an entire function.
Teorema      
A polynomial function is analytic at every point of C , i.e. is an entire function.
A rational function is analytic at each point of its domain.

2. Example Let , i.e. (i.e. and ). Then:
C-R equations mean here , i.e.
Thus, the origin is the only point where f can be derivable and f is nowhere analytic.

3. Example
If , the function f is analytic at every point of
With we have
Thus:
Cauchy-Riemann equations are verified by the first partial derivatives, and these derivatives are continuous functions on their domain. The function is differentiable at every point of its domain
The function f is analytic at every point of

4. Example
Let , i.e and
We have:
Thus, Cauchy-Rieman equations are verified if, and only if,
, i.e.
. It follows that
can be derivable only
on the line whose equation is
. For any point
on this line, every non empty open ball centered at
has points out of the line, therefore
cannot be
analytic anywhere (see Figure ).

5. Figure: Why a function is nowhere analytic