Difference between revisions of "Schwarz function"

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=Properties=
 
=Properties=
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[[Schwarz function is continuous]]<br />
<strong>Theorem:</strong> Let $M>0$. The Schwarz function is [[continuous]] on $(0,M)$.
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[[Schwarz function is nowhere differentiable on a dense subset]]<br />
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<strong>Proof:</strong>
 
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<strong>Theorem:</strong> Let $M>0$. The Schwarz function is [[nowhere differentiable]] on a [[dense]] subset of $(0,M)$.
 
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<strong>Proof:</strong> █
 
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=References=
 
=References=
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]<br />
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* {{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