Schwarz function

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