Difference between revisions of "Darboux function"
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$$D(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin\left((k+1)!x\right)}{k!},$$ | $$D(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin\left((k+1)!x\right)}{k!},$$ | ||
where $\sin$ denotes the [[sine]] function. | where $\sin$ denotes the [[sine]] function. | ||
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| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Darbouxplot.png|Plot of $D(x)$ on $[0,5]$. | ||
| + | </gallery> | ||
| + | </div> | ||
=Properties= | =Properties= | ||
Revision as of 18:27, 21 January 2016
The Darboux function is defined by $$D(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin\left((k+1)!x\right)}{k!},$$ where $\sin$ denotes the sine function.
Plot of $D(x)$ on $[0,5]$.
Properties
Theorem: The Darboux function is continuous on $\mathbb{R}$.
Proof: █
Theorem: The Darboux function is nowhere differentiable on $\mathbb{R}$.
Proof: █
