Difference between revisions of "Faber F1"
From specialfunctionswiki
| Line 1: | Line 1: | ||
The Faber function $F_1$ is defined by | The Faber function $F_1$ is defined by | ||
$$F_1(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{10^k} \displaystyle\inf_{m \in \mathbb{Z}} \left|2^{k!} x -m \right|.$$ | $$F_1(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{10^k} \displaystyle\inf_{m \in \mathbb{Z}} \left|2^{k!} x -m \right|.$$ | ||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Faberf1plot.png|Plot of $F_1$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The Faber function $F_1$ is [[continuous]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The Faber function $F_1$ is [[nowhere differentiable]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
=See Also= | =See Also= | ||
Revision as of 19:23, 22 January 2016
The Faber function $F_1$ is defined by $$F_1(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{10^k} \displaystyle\inf_{m \in \mathbb{Z}} \left|2^{k!} x -m \right|.$$
Plot of $F_1$.
Properties
Theorem: The Faber function $F_1$ is continuous.
Proof: █
Theorem: The Faber function $F_1$ is nowhere differentiable.
Proof: █
