View source for Complex Analysis/Differences from real differentiability

== n-times Real Differentiability ==

The function
: <math> f:\mathbb{R} \rightarrow \mathbb{R},</math> with <math>x \mapsto f(x)= |x|\cdot x</math>,
can be differentiated once. However, its first derivative is no longer differentiable at 0.

== Task ==

*Sketch the graphs of the functions <math>f</math> and <math>f'</math>.

*Can the function <math>f:\mathbb{R} \rightarrow \mathbb{R}</math> be extended to a holomorphic function <math>F:\mathbb{C} \rightarrow \mathbb{C}</math>, where <math>F_{|\mathbb{R}}=f</math> (i.e., <math>f(x)=F(x)</math> for all <math>x \in \mathbb{R}</math>)? Justify your answer using the properties of holomorphic functions!

*Show that the function
: <math> f_n:\mathbb{R} \rightarrow \mathbb{R},</math> with <math>x \mapsto f(x)= |x|\cdot x^n</math>,
: can be differentiated <math>n</math> times. However, the <math>n+1</math>-th derivative is no longer differentiable at 0.
== Remark ==
In complex analysis ([[Complex Analysis|Complex Analysis]]), one will see that a holomorphic function <math>f:U\to \mathbb{C}</math> defined on <math>U\subseteq \mathbb{C}</math> is automatically infinitely often complex differentiable if it is complex differentiable once (see[[w:/Criteria/|Holomorphy Criteria]].
== See also ==
*[[Complex Analysis|Complex Analysis]]
=== Translation and Version Control ===
This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Unterschiede _zur_reellen_Differenzierbarkeit
Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Kurs:Funktionentheorie/Unterschiede zur reellen Differenzierbarkeit|Kurs:Funktionentheorie/Unterschiede zur reellen Differenzierbarkeit]] - URL:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Unterschiede zur reellen Differenzierbarkeit
* Date: 12/17/2024

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[[de:Kurs:Funktionentheorie/Unterschiede zur reellen Differenzierbarkeit]]
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[[Category:Wiki2Reveal]]
[[Category:Complex Analysis]]