Theorems of Riemann

Theorems of Riemann primarily relate to the properties of functions and integrals. Riemann's Fundamental Theorem states that if a function is continuous, then its definite integral (area under the curve) can be found using an antiderivative. The Riemann integrability theorem states that a function is integrable if its set of discontinuities is small (like at only a few points). These theorems collectively formalize how we can approximate and calculate areas under curves, ensuring that for many functions, integration is a well-defined and reliable process.