Difference between revisions of "Petr function"
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=Properties= | =Properties= | ||
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| − | <strong>Theorem:</strong> The [[Petr function]] is [[continuous]]. | + | <strong>Theorem:</strong> The [[Petr function]] is [[continuous]] on $(0,1)$. |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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| − | <strong>Theorem:</strong> The [[Petr function]] is [[nowhere differentiable]]. | + | <strong>Theorem:</strong> The [[Petr function]] is [[nowhere differentiable]] on $(0,1)$. |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 12:14, 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 on $(0,1)$.
Proof: █
Theorem: The Petr function is nowhere differentiable on $(0,1)$.
Proof: █
