Difference between revisions of "Petr function"

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(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]...")
 
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c_{k-1} &; \mathrm{otherwise}.
 
c_{k-1} &; \mathrm{otherwise}.
 
\end{array} \right.$$
 
\end{array} \right.$$
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=Properties=
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<strong>Theorem:</strong> The [[Petr function]] is [[continuous]].
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<div class="mw-collapsible-content">
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<strong>Proof:</strong> █
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</div>
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<div class="toccolours mw-collapsible mw-collapsed">
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<strong>Theorem:</strong> The [[Petr function]] is [[nowhere differentiable]].
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<strong>Proof:</strong> █
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</div>
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</div>
  
 
=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]

Revision as of 12:13, 5 January 2016

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

Theorem: The Petr function is continuous.

Proof:

Theorem: The Petr function is nowhere differentiable.

Proof:

References

[1]