Archimedes used the method of exhaustion to calculate the area within a circle, finding the area of regular polygons with more and more sides. This was an early but informal example of the limit, one of the most basic concepts in mathematical analysis.

Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas go back to earlier mathematicians. The first results of analysis were implicitly present at the dawn of ancient Greek mathematics. For example, an infinite geometric sum is implicit in Zeno’s dichotomy paradox. Later Greek mathematicians such as Eudoxus and Archimedes made a more explicit but informal use of the concepts of limits and convergence when they used the . exhaustion method to calculate the area and volume of areas and solids. Explicit use of infinitesimals appears in Archimedes’ Method of Mechanical Theorems, a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chunzhi developed a method that would later be called the Cavalieri principle to determine the volume of a sphere in the 5th century. The Indian mathematician Bhaskara II gave examples of the derivative and used what is now known as Rolle’s theorem in the 12th century.

In the 14th century, Madhava of Sangamagrama developed an infinite series of expansions, such as the power series and the Taylor series, of functions such as sine, cosine, tangent and arctangent. In addition to developing the Taylor series of trigonometric functions, he also estimated the magnitude of the errors arising from truncation of these series and gave a rational approximation of the infinite series. His followers in the Kerala school of astronomy and mathematics extended his work into the 16th century.

The modern foundations of mathematical analysis were laid in 17th-century Europe. Descartes and Fermat independently developed analytic geometry, and a few decades later Newton and Leibniz independently developed the calculus of infinitesimals, which, through the stimulus of applied work that continued into the 18th century, evolved into such analytic topics as the calculus of variations, ordinary and partial derivative equations, Fourier analysis and derivative functions. During this period, calculus methods were used to approximate discrete problems by continuous ones.

In the 18th century, Euler introduced the concept of the mathematical function. Real analysis began to become a subject in its own right when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano’s work did not become widely known until the 1870s. In 1821, Cauchy began to put the calculus on a firm logical footing, rejecting the principle of generality of algebra that had been widely used in earlier works, notably by Euler. Instead, Cauchy formulated the calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required that an infinitesimal change in x corresponded to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence and began the formal theory of complex analysis. Poisson, Liouville, Fourier, and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of the limit approach, thus beginning the modern field of mathematical analysis.

In the middle of the 19th century, Riemann introduced his theory of integration. The last third of the century was marked by the arithmetization of analysis by Weierstrass, who believed that geometric reasoning was initially misleading, and introduced the definition of “epsilon-delta” from limit. Mathematicians then began to worry that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed real numbers using Dedekind’s abbreviations, in which irrational numbers are formally defined, which serve to fill the “gaps” between rational numbers, thus creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. Around that time, attempts to refine theorems from Riemann integration led to a study of the “size” of the set of discontinuities of real functions.

Also, “monsters” (nowhere continuous functions, continuous but nowhere differentiable functions, curves that fill space ) began to be investigated. In this context, Jordan developed his measure theory, Cantor developed what is now called naive set theory, and Baer proved Baer’s category theorem. In the early 20th century, calculus was formalized with axiomatic set theory. Lebeg solved the measure problem, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of a normalized vector space was in the air, and in the 1920s Banach created functional analysis.