Difference between revisions of "Thomae function"

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Thomae's function (sometimes called the popcorn function, raindrop function, Stars over Babylon) is given by the formula
 
Thomae's function (sometimes called the popcorn function, raindrop function, Stars over Babylon) is given by the formula
 
$$f(x) =\begin{cases}
 
$$f(x) =\begin{cases}
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=Properties=
 
=Properties=
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[[Thomae function is continuous at irrationals]]<br />
<strong>Theorem:</strong> The [[Thomae function]] is [[continuous]] at all [[irrational number|irrational numbers]] and discontinuous at all [[rational number|rational numbers]].
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[[Thomae function is discontinuous at rationals]]<br />
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The [[Thomae function]] has a (strict) [[local maximum]] at each [[rational number]].
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The [[Thomae function]] $f(x)$ is [[Riemann integral|Riemann integrable]] and
 
$$\displaystyle\int_0^1 f(x) dx = 0.$$
 
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<strong>Proof:</strong> █
 
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=Videos=
 
=Videos=
[https://www.youtube.com/watch?v=HeIU5lLtHyQ Thomae Function by Douglas Harder]<br />
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[https://www.youtube.com/watch?v=Xu5Y6DqzN7Q Thomae Function by Bret Benesh (11 January 2012)]<br />
[https://www.youtube.com/watch?v=Xu5Y6DqzN7Q Thomae Function by Bret Benesh]
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[https://www.youtube.com/watch?v=HeIU5lLtHyQ Thomae Function by Douglas Harder (19 April 2012)]<br />
  
 
=See also=
 
=See also=

Latest revision as of 00:36, 9 December 2016

Thomae's function (sometimes called the popcorn function, raindrop function, Stars over Babylon) is given by the formula $$f(x) =\begin{cases} 1 & \text{if } x= 0 \\ \tfrac1{q} & \text{if } x = \tfrac{p}{q}\\ 0 & \text{if } x \in \mathbb{R}-\mathbb{Q}. \end{cases}$$


Properties

Thomae function is continuous at irrationals
Thomae function is discontinuous at rationals

Videos

Thomae Function by Bret Benesh (11 January 2012)
Thomae Function by Douglas Harder (19 April 2012)

See also

Modifications of Thomae's Function and Differentiability

References

[1]
[2]
[3]