Complex Analysis, Complex Differentiable Question

[BrainHurts]

Homework Statement

Define f : \mathbb{C} \rightarrow \mathbb{C} by

f(z) = \left <br /> \{ <br /> \begin{array}{11} <br /> |z|^2 \sin (\frac{1}{|z|}), \mbox{when $z \ne 0$}, \\<br /> <br /> 0, \mbox{when z = 0} .<br /> \end{array}<br /> \right.<br />

Show that f is complex-differentiable at the origin although the partial derivative u_x is not continuous at origin.

Homework Equations

The Attempt at a Solution

To show that f is complex differentiable by defintion? In other words

f&#039;(0) = \lim_{h \rightarrow 0} \frac{f(0+h) - f(0)}{h} = \lim_{h \rightarrow 0} \frac{f(h) - 0}{h} = \lim_{h \rightarrow 0} \frac{|h|^2 \sin(\frac{1}{|h|})}{h} = \lim_{h \rightarrow 0} \frac{h\bar{h} \sin(\frac{1}{|h|})}{h} = 0 ?

Or am I missing something with \bar{h}. Because I'm assuming as h approaches 0, so does \bar{h}

Also, I see that

u_x(0,0) = \lim_{x \rightarrow 0} \frac{u(x,0)}{x}

A little help here, not sure how to approach this problem.

 

[BruceW]

Or am I missing something with \bar{h}. Because I'm assuming as h approaches 0, so does \bar{h}

why not write out h in polar form? and see what that implies for $\bar{h}$ as h goes to zero