Difference between revisions of "Thomae function"

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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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Revision as of 00:34, 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

Theorem: The Thomae function has a (strict) local maximum at each rational number.

Proof:

Theorem: The Thomae function $f(x)$ is Riemann integrable and $$\displaystyle\int_0^1 f(x) \mathrm{d}x = 0.$$

Proof:

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]