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$f(z)$ is analytic in all of $\Bbb C$
Let $f: \Bbb C \rightarrow \Bbb C$ be the function defined by
$$f(z) = \begin{cases} z \; \; \text{if} \; \; |z| \leq 1 \\ z^2 \; \; \text{if} \; \; |z| > 1 \end{cases}$$
Show that $f$ is analytic in all of $\Bbb C$
I can’t see how to get started on this. Can someone please help. Thank you.
A:
$f$ is analytic at all points of the open unit disc, so $f$ is analytic on the open unit disc. Now $f$ is of the form $z\mapsto z^k$ for some integer $k\in\mathbb N$. We note that $f$ is even, so $f$ must be analytic on $\mathbb C$. The limit definition of the derivative is only true for one-sided limits, so in general $f$ cannot be differentiable on the entire line.
Rollover, Missouri
Rollover is an unincorporated community in Barry County, Missouri, United States. It is located on Missouri Route D, approximately 7.5 miles north of Ozark.
History
A post office called Rollover was established in 1877, and remained in operation until 1922. The name Rollover is said to have been chosen from a nearby hill that resembled a ball lying on its side.
References
Category:Unincorporated communities in Barry County, Missouri
Category:Unincorporated communities in MissouriSaturday, April 30, 2016
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