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Real Analysis: Definition

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ordinary derivative of a constant function

Derivative formulas work–but why do they work?

Real analysis is the rigorous version of calculus (“analysis” is the branch of mathematics that deals with inequalities and limits). It’s an extension of calculus with new concepts and techniques of proof (Bloch, 2011), filling the gaps left in an introductory calculus class (Trench, 2013). Those “gaps” are the pure math underlying the concepts of limits, derivatives and integrals. Instead of just using those formulas, you’ll be proving that they work.

Essentially, you go from the “what” in calculus is true to “why” it’s true (Lebl, 2020).

What is Taught in Real Analysis?

A typical real analysis course includes:

Real analysis heavily emphasizes proofs. In fact, you can barely take a step in the course without axioms and postulates: every formula encountered in the class has to be proved. Instead of using formulas for derivatives and integration, you’ll be proving that they work.

What Are Some Uses for Real Analysis?

Real analysis underpins many other branches of mathematics including probability theory and analytic number theory. However, it isn’t just a scholarly exercise in pedantry. Real world uses include:


Lebl, J. (2020). Basic Analysis I. Retrieved September 20, 2020 from:
Bloch, E. (2011). The Real Numbers and RA. Springer.
Schramm, M. (2012). Introduction to RA. Dover Publications.
MCMullen, C. Course Notes. Retrieved September 20, 2020 from:
Trench, W. (2013). Introduction to RA (TRENCH_REAL_ANALYSIS).

Stephanie Glen. "Real Analysis: Definition" From Calculus for the rest of us!

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