Difference between revisions of "Schwarz function"
From specialfunctionswiki
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| − | [[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= | ||
Revision as of 17:46, 20 September 2016
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
