Скачать или смотреть The Dirichlet Function: A Pathological Counterexample

The provided source offers an overview of the *Dirichlet function**, a mathematical concept named after Peter Gustav Lejeune Dirichlet, which serves as an **indicator function for rational numbers within the set of real numbers**. This function outputs 1 if the input is rational and 0 if it's irrational. Notably, it is a **pathological function**, meaning it exhibits counterintuitive properties, such as being **nowhere continuous* and *not Riemann-integrable* on any segment despite being bounded. However, the Dirichlet function **is Lebesgue-integrable**, with an integral of zero over the real numbers. It also demonstrates a unique **periodicity**, where its set of periods is the dense set of rational numbers.