Difference between revisions of "Petr function"
From specialfunctionswiki
(Created page with "Let $x \in [0,1]$ have decimal representation $x=\displaystyle\sum_{k=1}^{\infty} \dfrac{a_k}{10^k}$, where $a_k \in \{0,1,\ldots,9\}$. The Petr function $P_K \colon [0,1]...") |
(→Properties) |
||
| (3 intermediate revisions by the same user not shown) | |||
| Line 6: | Line 6: | ||
c_{k-1} &; \mathrm{otherwise}. | c_{k-1} &; \mathrm{otherwise}. | ||
\end{array} \right.$$ | \end{array} \right.$$ | ||
| + | |||
| + | =Properties= | ||
| + | [[The Petr function is continuous]]<br /> | ||
| + | [[The Petr function is nowhere differentiable]]<br /> | ||
=References= | =References= | ||
[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] | [https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf] | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 20:34, 25 June 2017
Let $x \in [0,1]$ have decimal representation $x=\displaystyle\sum_{k=1}^{\infty} \dfrac{a_k}{10^k}$, where $a_k \in \{0,1,\ldots,9\}$. The Petr function $P_K \colon [0,1] \rightarrow \mathbb{R}$ is defined by $$P_K(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{c_k b_k}{2^k},$$ where $b_k = a_k \mod 2, c_1=1$, and for $k \geq 2$, $$c_k = \left\{ \begin{array}{ll} -c_{k-1} &; a_{k-1} \in \{1,3,5,7\}, \\ c_{k-1} &; \mathrm{otherwise}. \end{array} \right.$$
Properties
The Petr function is continuous
The Petr function is nowhere differentiable
