I've recently started complex analysis but I have very little background in complex numbers and to make sure I don't fall behind I'm doing some extra exercises one of which is
Show $f$ is continuous on $\mathbb C$
$${f(z)=\begin{cases}
z^{2} / |z| \qquad &\mbox{when } z\neq0,\\ 0
\qquad &\mbox{when }z=0.
\end{cases}}$$
I know the epsilon delta proof for limits quite well and the limit definition of continuous but I don't know how to apply them to this function. Can anybody help or at least get me started?
Best Answer
I'm sure you will agree with me that for $z \not= 0$, $f(z)$ is continuous.
The only point of concern is the continuity at $z=0$.
Now consider
$$|f(z)|=|\frac{z^2}{|z|}|=\frac{|z|^2}{|z|}=|z|$$
and we can use a result:
$|f|$ has a limit $0$ as $z \rightarrow c$ $\iff$ $f$ has a limit $0$ as $z \rightarrow c$
And we have
$$\lim_{z \to 0}|z|=0$$ So $$\lim_{z \to 0}f(z)=0=f(0)$$
and it follows that $f$ is continuous at $z=0$.