Difference between revisions of "Schwarz function"
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=Properties= | =Properties= | ||
| − | [[Schwarz function is continuous]] | + | [[Schwarz function is continuous]]<br /> |
| − | [[Schwarz function is nowhere differentiable on a dense subset]] | + | [[Schwarz function is nowhere differentiable on a dense subset]]<br /> |
=References= | =References= | ||
| − | + | * {{BookReference|Continuous Nowhere Differentiable Functions|2003|Johan Thim|prev=Darboux function|next=findme}} $\S 3.5$, pg. 28 | |
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Latest revision as of 18:03, 25 June 2017
Define $\varphi(x)=\lfloor x \rfloor + \sqrt{x-\lfloor x \rfloor}$, where $\lfloor \cdot \rfloor$ denotes the floor function and let $M>0$. The Schwarz function $S \colon (0,M) \rightarrow \mathbb{R}$ is defined by $$S(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\varphi(2^k x)}{4^k}.$$
Properties
Schwarz function is continuous
Schwarz function is nowhere differentiable on a dense subset
References
- 2003: Johan Thim: Continuous Nowhere Differentiable Functions ... (previous) ... (next) $\S 3.5$, pg. 28
