Reaalanalüüsi

Reaalanalüüsi, often translated as real analysis, is a branch of mathematical analysis that studies the behavior of real numbers, sequences, and series, and the properties of functions defined on the set of real numbers. It provides the rigorous foundation for calculus, extending its concepts to more general settings and establishing precise definitions and proofs for fundamental theorems. Key topics within real analysis include the properties of the real number system itself, such as completeness and its topological characteristics. It delves deeply into the theory of limits, continuity, and differentiability of functions. A central theme is the concept of convergence, applied to sequences of numbers and infinite series. The integral calculus, particularly the Riemann integral and its extensions, is a significant area of study, along with measure theory and Lebesgue integration, which offer a more powerful framework for integration. Real analysis also examines uniform convergence, Fourier series, and the properties of function spaces. Its methods and results are crucial for many other areas of mathematics, physics, engineering, and economics.