Difference between revisions of "McCarthy function"

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=Properties=
 
=Properties=
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[[McCarthy function is continuous]]
<strong>Theorem:</strong> The [[McCarthy function]] is [[continuous]] on $\mathbb{R}$.
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[[McCarthy function is nowhere differentiable]]
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The [[McCarthy function]] is [[nowhere differentiable]] on $\mathbb{R}$.
 
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<strong>Proof:</strong> █
 
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=References=
 
=References=

Revision as of 13:39, 17 November 2016

The McCarthy function $M$ is defined by $$M(x) = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{2^k} g \left( 2^{2^k} x \right),$$ where $$g(x) = \left\{ \begin{array}{ll} 1+x &; x \in [-2,0] \\ 1-x &; x \in [0,2], \end{array} \right.$$ and $g(x+4)=g(x)$ for any $x \in \mathbb{R}$.

Properties

McCarthy function is continuous McCarthy function is nowhere differentiable

References

[1]