Thomae function

From specialfunctionswiki
Revision as of 21:28, 4 July 2016 by Tom (talk | contribs)
Jump to: navigation, search

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

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 Riemann integrable and $$\displaystyle\int_0^1 f(x) \mathrm{d}x = 0.$$

Proof:

Videos

Thomae Function by Douglas Harder
Thomae Function by Bret Benesh

See also

Modifications of Thomae's Function and Differentiability

References

[1]
[2]
[3]