Complex Analysis/Differences from real differentiability

n-times Real Differentiability

The function

${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} ,}$ with ${\displaystyle x\mapsto f(x)=|x|\cdot x}$,

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

Task

• Sketch the graphs of the functions ${\displaystyle f}$ and ${\displaystyle f'}$.

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

• Show that the function

${\displaystyle f_{n}:\mathbb {R} \rightarrow \mathbb {R} ,}$ with ${\displaystyle x\mapsto f(x)=|x|\cdot x^{n}}$,

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

Remark

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

See also

• Complex Analysis