Difference between revisions of "Darboux function"
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| − | <strong>Theorem:</strong> The Darboux function is [[continuous]]. | + | <strong>Theorem:</strong> The Darboux function is [[continuous]] on $\mathbb{R}$. |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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| − | <strong>Theorem:</strong> The Darboux function is [[nowhere differentiable]]. | + | <strong>Theorem:</strong> The Darboux function is [[nowhere differentiable]] on $\mathbb{R}$. |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 22:56, 31 December 2015
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.
Properties
Theorem: The Darboux function is continuous on $\mathbb{R}$.
Proof: █
Theorem: The Darboux function is nowhere differentiable on $\mathbb{R}$.
Proof: █
