Difference between revisions of "Thomae function"
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=Properties= | =Properties= | ||
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| − | <strong>Theorem:</strong> The Thomae function is [[continuous]] at all [[irrational number|irrational numbers]] and discontinuous at all [[rational number|rational numbers]]. | + | <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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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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| − | <strong>Theorem:</strong> The Thomae function has a (strict) [[local maximum]] at each [[rational number]]. | + | <strong>Theorem:</strong> The [[Thomae function]] has a (strict) [[local maximum]] at each [[rational number]]. |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 20:49, 11 April 2015
Thomae's function is given by the formula $$f(x) =\begin{cases} 1 & \text{if } x= 0 \\ \tfrac1{q} & \text{if } x = \tfrac{p}{q}\\ 0 & \text{if } x \in \mathbb{R}-\mathbb{Q}. \end{cases}$$
Plot of the Thomae function.
Properties
Theorem: The Thomae function is continuous at all irrational numbers and discontinuous at all rational numbers.
Proof: █
Theorem: The Thomae function has a (strict) local maximum at each rational number.
Proof: █
Theorem: The Thomae function $f(x)$ is not Riemann integrable but it is Lebesgue integrable and $$\displaystyle\int_0^1 f(x) dx = 0.$$
Proof: █
