Thomaefüggvény

Thomaefüggvény, also known as Thomae's function or the popcorn function, is a classical example in real analysis illustrating a function that is continuous at every irrational point and discontinuous at every rational point. It is named after the 19th‑century German mathematician Johannes Thomae and is frequently cited for its simple definition and interesting convergence properties.

Definition: For x ∈ R, if x is irrational then f(x) = 0. If x is rational, write x = p/q in lowest terms with integers p, q and q > 0, gcd(p, q) = 1, and set f(x) = 1/q. In particular, since 0 = 0/1, one has f(0) = 1.

Properties: The range of the function is {0} ∪ {1/q : q ∈ N}. The function is bounded between 0 and 1. Thomaefüggvény is discontinuous at every rational and continuous at every irrational; since the rationals are dense in R, the set of discontinuities is dense, while the set of continuity points (the irrationals) has full measure zero for the set of nonzero values.

Integrability: On any closed interval, Thomaefüggvény is Riemann integrable and its integral is 0. The Lebesgue integral is also 0 because the function is zero almost everywhere; the nonzero values occur on the rationals, which form a set of measure zero.

Variants and usage: It is commonly studied on [0,1], with extensions to all of R by the same rule or by periodic extension. The graph is often called the popcorn graph due to its spike-like shape.

See also: Popcorn function, Dirichlet function. Further reading includes standard texts in real analysis discussing examples of continuity and integrability.