Real Analysis
Real analysis is one of the core branches of mathematics. It is the study of the properties of real numbers, sequences, and functions.
Due to its foundational nature, real analysis is often taught in a very rigorous manner, with a focus on theorems, proofs, and definitions. However, this can make it seem intimidating to beginners. As a result, many students struggle to see the big picture and the intuition behind the concepts.
Hence, we aim to provide a gentle introduction to real analysis, focusing on the key ideas and intuitions along with motivating examples. We will cover topics such as limits, continuity, differentiation, and integration, building up the theory step by step. Of course, this is much harder than it sounds.
What Exactly is Real Analysis?
Let's begin by understanding what real analysis is all about. It has two main components - real and analysis.
Analysis in a sense is the study of functions. It deals with the properties of functions, their behavior, and their relationships with other functions. In particular, it focuses on the concepts of limits, continuity, differentiation, and integration. Hearing these terms might remind you of calculus, but there is a key difference. While in Calculus we often focus on the intuition and applications of these concepts, in Analysis we delve deeper into the theory behind them. For example, when studying integration, we can rigorously prove the existence of definite integrals and explore the properties of integrable functions.
Real refers to the real numbers. This means that in real analysis, we are primarily concerned with functions defined on the real number line. This also means that there are other types of analysis, such as complex analysis, which deal with functions defined on the complex plane.
So, in essence, real analysis is the study of functions defined on the real numbers, with a focus on the theoretical underpinnings of concepts like limits, continuity, differentiation, and integration.