Thomaes function is a real valued function of a real variable, named after Carl Johannes Thomae, defined to equal 1 over q at every rational number p over q written in lowest terms, and to equal 0 at every irrational number. Thomae originally introduced it in an early textbook on Riemanns theory of integration as an example of a function that is integrable despite having infinitely many discontinuities. The function is discontinuous at every rational number but continuous at every irrational number, so its points of discontinuity and its points of continuity are both dense in the real numbers, a striking and counterintuitive property that makes it a standard teaching example in real analysis. It is also known by several informal names describing its graph, including the popcorn function, the raindrop function, and the modified Dirichlet function, and the mathematician John Horton Conway called it Stars over Babylon.
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1. Thomae's Function (Wikipedia)
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Thomae's function is a real-valued function of a real variable that can be defined as: f ( x ) = { 1 q if x = p q ( x is rational), with p ∈ Z and q ∈ N coprime 0 if x is irrational. {\displaystyle f(x)={\begin{cases}{\frac {1}{q}}&{\text{if }}x={\tfrac {p}{q}}\quad (x{\text{ is rational)
Principal Result Of: Carl Johannes Thomae, Lead
It is named after Carl Johannes Thomae, but has many other names
Attributed To: Carl Johannes Thomae, Lead sentence
Thomae's function is a real-valued function of a real variable that can be defined as: f ( x ) = { 1 q if x = p q ( x is rational), with p ∈ Z and q ∈ N coprime 0 if x is irrational. {\displaystyle f(x)={\begin{cases}{\frac {1}{q}}&{\text{if }}x={\tfrac {p}{q}}\quad (x{\text{ is rational), with }}p\in \mathbb {Z} {\text{ and }}q\in \mathbb {N} {\text{ coprime}}\\0&{\text{if }}x{\text{ is irrational.}}\end{cases}}} It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function (not to be confused with the integer ruler function), the Riemann function, or the Stars over Babylon (John Horton Conway's name).
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