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A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible.
@pie, the greatest commen divisor of $0$ and $1$ is $1$. So in terms of the definition of Thomae function we have $0=\frac {0} {1}$. Hence, $t (0)=\frac {1} {1}=1$.
Aug 17, 2023 · Thomae Function is Continuous at Irrational Numbers This article was Featured Proof between 17th December 2020 and 12th September 2021.
Oct 27, 2025 · This article examines the Thomae function, a paradigmatic exam-ple of a function that is continuous on the irrationals and discontinuous elsewhere. Defined for a parameter θ > 0, it exhibits a rich self...
Thomae’s function is a function that maps irrational numbers to positive rational numbers and zero to zero. This paper shows that Thomae’s function is continuous at every irrational point using a lemma and an argument...
Since there are not obvious candidate functions which satisfy the property we are looking for, we are going to need to be more systematic and rigorous in our quest.
The function is a modified version of the Dirichlet function: usually defined as 1 for rational numbers (i.e. those numbers that can be written as a ratio, like 3/4 or 10/5) and 0 elsewhere.
Throughout this chapter, A is a non-empty subset of R and f: A → R is a function. The function f is continuous at c ∈ A if for any given ε> 0 there exists δ> 0 such that if x ∈ A and | x c | <δ then | f (x) f (c) | <ε...
Some Topics Covered Continuous functions are integrable, bounded and continuous at all but finitely many points; Thomae's Function.
Thomae's function is a real-valued function of a real variable that can be defined as: f (x)= {1qif x=pq (x is rational), with p∈ℤ and q∈ℕ coprime0if x is irrational. It is named after Carl Johannes Thomae, but has ma...
