The simplest open neighborhood of a point
is an open ball centered at
; we will generally use such balls.
Definition 3.5.2
Let
be a function defined over an open subset
of
. the function
is
analytic over
if it is analytic at each point of
.
A function which is analytic over the whole of
is called an entire function.
Proposition 3.5.3
 A polynomial function is analytic at every point of
, i.e. is an entire function.
 A rational function is analytic at each point of its domain.
Proof.
 A polynomial function is an entire function, as a consequence of Proposition 2.4 and Ex. 2.3.
 We use once again Proposition 2.4, together with the previous alinea,
as a rational function is the quotient of two polynomial functions.
Example 3.5.4
Let
, i.e.
(i.e.
and
). Then:
CR equations mean here
,
i.e.
. Thus, the origin is the only point where
can
be derivable and
is nowhere analytic.
Example 3.5.5
We use the function and the results of Example
3.5.
For any point
on one of these axes, any open ball centered at
contains points where
is not differentiable. Thus,
is not analytic at any point.
Example 3.5.6
If
, the function
is analytic at every
point of
:
 With the notations of 3.1, we have:
Thus:
 CauchyRiemann equations are verified by the first partial
derivatives, and these derivatives are continuous functions on their
domain. By Thm 3.4, the function
is
differentiable at every point of its domain
.
 As
is not defined at
and at
,
is not entire.
Example 3.5.7
Let
, i.e.
and
. We have:
Thus, CauchyRieman 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
3).
Figure 3:
Why a function is nowhere analytic.

Noah DanaPicard
20071224