Complex Analysis/Differences from real differentiability

n-times Real Differentiability

The function

$f:ℝ\to ℝ,$ with $x\mapsto f(x)=|x|\cdot x$,

can be differentiated once. However, its first derivative is no longer differentiable at 0.

Task

• Sketch the graphs of the functions $f$ and $f^{\prime }$.

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

• Show that the function

$f_n:ℝ\to ℝ,$ with $x\mapsto f(x)=|x|\cdot x^n$,

can be differentiated $n$ times. However, the $n+1$-th derivative is no longer differentiable at 0.

Remark

In complex analysis (Complex Analysis), one will see that a holomorphic function $f:U\to ℂ$ defined on $U\subseteq ℂ$ is automatically infinitely often complex differentiable if it is complex differentiable once (seeHolomorphy Criteria.

See also

• Complex Analysis

Translation and Version Control

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• Date: 12/17/2024

de:Kurs:Funktionentheorie/Unterschiede zur reellen Differenzierbarkeit