Introduction
Real analysis is the rigorous branch of mathematics that explores real numbers, sequences, series, and real-valued functions. It delves into foundational concepts such as convergence, limits, continuity, differentiability, and integration—ensuring mathematical reasoning is sound and precise. Central to the field is the completeness of the real number system (e.g., the least upper bound property), which underpins many essential theorems like the intermediate value theorem and the mean value theorem. Beyond serving as a theoretical backbone to calculus, real analysis forms the logical framework for advanced topics in topology, functional analysis, and applied sciences.
Contents
Single-variable
Number Systems (This might be too much for the sake of real analysis, but we’d better know the basics of real numbers before really getting into its “analysis” 🙂)
Sequences
Real Continuous Functions
Differentiation of Real Functions
Integral
Series
Multi-variable
Directional Derivative, Gradient and Jacobian
Matrix and Vector Calculus
Multivariable Integration
Ordinary Differential Equations
More …
After developing the theory of real analysis, people extend the theory of integration/differentiation in the following ways:
Complex Analysis - Obviously, one might ask, what about the complex case?
Differential Geometry - This is the geometrical way to extend real analysis and multi-variable calculus. It gives rigorous geometrical meaning to directional derivatives (tangent vectors), differentiation (differential forms) and integrals (volume);
Measure Theory - Using the tool of measures, one can develop a nicer, and more abstract theory of integration, called the Lebesgue integral, which addresses some deficiencies of Riemann integration:
- Riemann integration does not handle functions with many discontinuities;
- Riemann integration does not handle unbounded functions;
- Riemann integration does not work well with limits.
Details about why Riemann integral is not good enough can be found in Section 1B, Axler.