**Example 3.9.** Show
that the function

is differentiable at points that lie on the -axis,
and at points that lie on the -axis,
but is
nowhere analytic.

**Explore Solution 3.9.**

**Solution.** Recall
Definition 3.1 (from Section
3.1): when we say a function is analytic at a
point
we mean that the function

is differentiable not only at ,
but also at every point in some neighborhood of . With
this in mind, we proceed

to determine where the Cauchy-Riemann
equations (3-16) are
satisfied. We write

and ,

and compute the partial derivatives:

, , and

, .

Here are
continuous, and

holds for all points
in the complex plane.

But if
and only if , which
is equivalent to

.

Hence, the Cauchy-Riemann
equations hold only at the
points where .

According to Theorem 3.4, is
differentiable only
when ,

which occurs only at points that lie on
the coordinate axes. Furthermore, for any point on the
coordinate axes,

there contains an -neighborhood
about it, in which there exist points where is
not differentiable.

Applying Definition 3.1 (from Section
3.1) , we see that the
function

is not analytic on either of the
coordinate axes.

Therefore, is
nowhere analytic.

**We are done.**

**Aside.** Both
and
can assist us in finding the partial derivatives.

**Aside.** The
*Mathematica* solution uses the commands.

**Aside.** The Maple
commands are similar.

**>**** **

**>**** **

**>**** **

**>**** **

**>**** **

**>**** **

**>**** **

Here are
continuous, and

holds for all points
in the complex plane.

But if
and only if , which
is equivalent to

.

Hence, the Cauchy-Riemann equations hold
only at the points
where .

According to Theorem 3.4, is
differentiable only
when ,

which occurs only at points that lie on
the coordinate axes. Furthermore, for any point on the
coordinate axes,

there contains an -neighborhood
about it, in which there exist points where is
not differentiable.

Applying Definition 3.1 (from Section
3.1) , we see that the
function

is not analytic on either of the
coordinate axes.

Therefore, is
nowhere analytic.

**Remark.** In this
book the use of computers is optional.

Hopefully this text will promote their use and understanding.

This solution is complements of the authors.

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2011 John H. Mathews, Russell W. Howell