Most of ancient and medieval mathematics concerned geometry and algebra. Questions of analysis arose in the work of Euler and Stirling and others, but only in isolated morsels. The real need for analysis became apparent when Newton and Leibniz's calculus took hold. This powerful new set of tools required some theoretical underpinning, some rigorous foundation, and analysis was the tool that was needed to carry out the program.

It was not until the nineteenth century that the necessary talent and focus came together to produce analysis as we know it today. Cauchy, Weierstrass, Riemann, and many others laid the foundations of the subject, provided the necessary definitions, and proved the required theorems. In the twentieth century Zygmund, Besicovitch, Hardy, Littlewood, and many others have carried the torch and continued to develop the subject.

The importance and centrality of real analysis is certainly confirmed by the fact that virtually every graduate programin the country—indeed, in the world—requires its students to take a qualifying exam in the subject. We are exposed to real analysis, both at the undergraduate and graduate levels. Today, analysis has assumed a newly prominent position in the infrastructure because of many new engineering applications such as wavelets, and also new financial applications such as the Black-Scholes theory of option pricing.

The fact remains that real analysis continues to be a rather technical and recondite subject. This is in part because mastery of the discipline is more a matter of technique than erudition or conceptual development.