Difference between revisions of "Schwarz function"
From specialfunctionswiki
(Created page with "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...") |
|||
| Line 19: | Line 19: | ||
=References= | =References= | ||
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]<br /> | [https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]<br /> | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Revision as of 18:35, 24 May 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
Theorem: Let $M>0$. The Schwarz function is continuous on $(0,M)$.
Proof: █
Theorem: Let $M>0$. The Schwarz function is nowhere differentiable on a dense subset of $(0,M)$.
Proof: █
