Difference between revisions of "Thomae function"
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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> █ | ||
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| + | <strong>Theorem:</strong> The [[Thomae function]] $f(x)$ is not [[Riemann integrable]] but it is [[Lebesgue integrable]] and | ||
| + | $$\displaystyle\int_0^1 f(x) dx = 0.$$ | ||
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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Revision as of 20:37, 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: █
