Dirichletfüggvény

The Dirichletfüggvény, known in English as the Dirichlet function, is a mathematical function named after the German mathematician Johann Peter Gustav Lejeune Dirichlet. It is primarily defined on the real number line and serves as a classical example of a function that is discontinuous at many points yet illustrates important concepts in real analysis and measure theory.

The function is typically defined as a characteristic function of the rational numbers: it takes the value 1 for rational numbers and 0 for irrational numbers. Formally, if Q denotes the set of rational numbers and R the set of real numbers, then the Dirichlet function f(x) is expressed as: f(x) = 1 if x is rational, and f(x) = 0 if x is irrational. This definition creates a function that is highly discontinuous, as at every point in the domain, the function exhibits a jump between 0 and 1 arbitrarily close to that point.

Despite being discontinuous everywhere, the Dirichlet function is significant in various mathematical contexts. It demonstrates that a function can be nowhere continuous, challenging intuitive notions of smoothness. The function is also a key example in measure theory, illustrating that the set of rational numbers has Lebesgue measure zero, making the function Lebesgue integrable with an integral value of zero.

The Dirichlet function underscores the difference between continuity properties and measure-theoretic concepts, serving as an essential example in understanding the rigorous foundations of modern analysis. Its properties highlight the complexity of functions defined via dense subsets of the real numbers and the importance of measure theory in analyzing functions with complicated discontinuity sets.