Real Analysis

What exactly are limits, continuity and the real numbers, and why did calculus need them defined?

5 chapters4 min read5 turning points 0 open problems

Branched from: Calculus + Fourier Analysis
Branched into: Continuous Optimisation + Ergodic Theory + Probability Theory + Set Theory + Stochastic Processes
Figures: Augustin-Louis Cauchy, Bernard Bolzano, Bernhard Riemann, Karl Weierstrass, Richard Dedekind, Georg Cantor, Henri Lebesgue

In brief

Real analysis is calculus rebuilt on precise definitions. Instead of infinitely small quantities, it uses limits, defined by a game of tolerances: for every accuracy you demand, there is a point beyond which the values stay that close. Continuity, derivatives, integrals and infinite series are all defined this way, and so, finally, are the real numbers themselves.

It was forced into existence by failures. Intuition said every continuous curve has a slope somewhere, that series of continuous functions stay continuous, and that every function worth studying has an integral. Each turned out to be false. What came out of the repair (rigorous proof, set theory, measure) became the foundation of modern mathematics.

Key ideas

Limit (ε–δ)Enters 1821: $\lim_{x \to a} f(x) = L$ means: for every $\varepsilon > 0$ there is a $\delta > 0$ such that $|f(x) - L| < \varepsilon$ whenever $0 < |x - a| < \delta$ . Infinitesimals are replaced by a precise promise.
ContinuityEnters 1872: A function is continuous if small changes in input give small changes in output, defined with limits. Continuity does not imply smoothness, as Weierstrass showed.
Completeness of the real numbersEnters 1872 – 1874: The real number line has no gaps: every bounded increasing sequence has a limit. Dedekind and Cantor constructed the reals from the rationals to guarantee it.
Uniform convergenceEnters 1821: A sequence of functions converging at the same rate everywhere, strong enough to guarantee that limits of continuous functions stay continuous. Ordinary pointwise convergence is not.
Measure and the Lebesgue integralEnters 1902: A way to assign a size to very general sets, and an integral built on it that handles far wilder functions and limits than Riemann's. It is the modern standard.

Chapter I

Answering Berkeley

For a century after Berkeley's attack, calculus ran on results rather than foundations. What changed that was teaching. At the École Polytechnique, Augustin-Louis Cauchy had to explain calculus to engineering students, and his Cours d'analyse (1821) tried to prove everything. It defined a limit as a value that quantities approach as closely as desired, a continuous function as one where small changes in input give small changes in output, and an infinite series as convergent when its partial sums settle down. The ghosts of departed quantities were replaced by statements about arbitrarily small, but always finite, numbers.

In Prague, the priest-philosopher Bernard Bolzano had reached similar definitions a few years earlier, in pamphlets almost no one read. Whether Cauchy saw them is still argued.

Chapter II

Fourier's Pressure

What made rigour urgent was Fourier analysis. Fourier series produced functions with jumps and corners as limits of smooth waves, contradicting a theorem in Cauchy's own book that limits of continuous functions are continuous. To say which functions have Fourier series, Riemann had to define the integral itself, in 1854. To say where a Fourier series could fail, Cantor was led to study strange infinite sets of points.

Then came the counterexamples. In 1872 Karl Weierstrass, who taught analysis in Berlin with ε and δ exactly as students learn it today, presented a function that is continuous everywhere and has a slope nowhere: a curve that is all corners. Charles Hermite wrote of turning away "with fright and horror from this lamentable plague of continuous functions which do not have derivatives". The lesson was blunt. Geometric intuition had misled the best mathematicians for two centuries, and only precise definitions could be trusted.

Chapter III

What Is a Real Number?

Limits need something to converge to. If the number line had gaps, a sequence could close in on a hole. Nobody had defined the real numbers; everyone had assumed them. In 1872 Richard Dedekind and Georg Cantor independently constructed them from the rationals, Dedekind by cutting the rational line into two pieces, Cantor by sequences that bunch together. Two years later Cantor proved that the real numbers cannot be listed one by one: there are strictly more of them than whole numbers. Infinity came in different sizes, and set theory was born.

Chapter IV

A Closer Look: When Intuition About Limits Fails

Cauchy's Cours d'analyse stated that a convergent series of continuous functions is continuous. Here is why that is false. Take the functions $f_n(x) = x^n$ on the interval $[0, 1]$ . Each is continuous, a smooth curve. As $n$ grows, for any $x < 1$ the values $x^n$ shrink to 0, while at $x = 1$ they stay at 1. The limit is

f(x) = \begin{cases} 0 & 0 \le x < 1, \\ 1 & x = 1, \end{cases}

which jumps. Continuous functions converged, point by point, to a discontinuous one.

The repair is uniform convergence: require that a single $n$ make $f_n$ close to $f$ at every $x$ at once. Here that fails. However large $n$ is, points just below 1 still have $x^n$ near 1, far from the limit 0. Seidel, Stokes and Weierstrass showed that uniform limits of continuous functions are continuous, and the distinction between the two kinds of convergence became basic analysis.

Precise definitions are what make such distinctions possible. The statement $\lim_{x \to 2} x^2 = 4$ means: for every tolerance $\varepsilon > 0$ there is a $\delta > 0$ such that $|x^2 - 4| < \varepsilon$ whenever $|x - 2| < \delta$ . To prove it, factor $|x^2 - 4| = |x - 2|\,|x + 2|$ . If $|x - 2| < 1$ then $|x + 2| < 5$ , so $|x^2 - 4| < 5|x - 2|$ . Choosing

\delta = \min\!\left(1, \frac{\varepsilon}{5}\right)

guarantees $|x^2 - 4| < \varepsilon$ . No infinitesimals are involved, only finite numbers and a promise that holds for every tolerance. That is the whole of Weierstrass's answer to Berkeley.

Chapter V

Measure

The last repair was to the integral. Riemann's integral cannot handle Dirichlet's function (1 on rationals, 0 on irrationals), and it behaves badly under limits. In 1902 Henri Lebesgue built a new one on a theory of measure, a consistent way of assigning size to very general sets. He compared it to counting money. Riemann adds up coins in the order he picks them up, while Lebesgue first sorts them by value. Measure theory became the foundation of modern analysis, and thirty years later the foundation of probability too.

Turning points5

1821REFORMULATIONContested
Cauchy's Cours d'analyse
Augustin-Louis Cauchy's textbook for the École Polytechnique defines limits, continuity and convergence in terms of quantities that become arbitrarily small, and proves theorems from those definitions. It is the first systematic attempt to answer Berkeley and ground calculus without infinitesimals. It still contained errors, including a false theorem that a convergent sum of continuous functions is continuous.
What is disputed
Bernard Bolzano, in Prague, had published similar definitions of continuity in 1817 in an obscure pamphlet. In 1970 the historian Ivor Grattan-Guinness argued that Cauchy had read and used Bolzano's work without acknowledgement. Others, including Hans Freudenthal, rejected the charge as unsupported. The question remains open. Bolzano is in any case recognised as an independent pioneer.
Augustin-Louis Cauchy, Bernard Bolzano
› Sources (3)
- Cauchy, A.-L. (1821). Cours d'analyse de l'École royale polytechnique. Paris.
- Grattan-Guinness, I. (1970). Bolzano, Cauchy and the 'new analysis' of the early nineteenth century. Archive for History of Exact Sciences 6: 372–400.
- Grabiner, J. V. (1981). The Origins of Cauchy's Rigorous Calculus. MIT Press.
1854 (published 1868)REFORMULATION
Riemann defines the integral
In his habilitation thesis on trigonometric series, Bernhard Riemann gives the first precise general definition of the integral, by limits of sums over finer and finer partitions. He needs it to decide which functions have Fourier coefficients at all. He also constructs a function that is discontinuous at a dense set of points yet still integrable.
Bernhard Riemann
› Sources (1)
- Riemann, B. (1868). Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13.
1872DISPROOF
A continuous curve with no slope anywhere
Karl Weierstrass presents to the Berlin Academy a function, a sum of cosine waves of rapidly increasing frequency, that is continuous everywhere and differentiable nowhere. It refutes the belief, held by nearly every mathematician, that a continuous curve must be smooth at most points. Hermite called such functions a "lamentable plague". Intuition could no longer be trusted, and only definitions could be.
Karl Weierstrass
› Sources (1)
- Weierstrass, K. (1872). Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen. Lecture to the Königlich Preussische Akademie der Wissenschaften, Berlin, 18 July 1872.
1872 – 1874REFORMULATION
The real numbers are constructed
In 1872 Richard Dedekind (by "cuts" of the rationals) and Georg Cantor (by convergent sequences) independently construct the real numbers, proving the gap-free property that calculus had always assumed. In 1874 Cantor shows that the real numbers cannot be listed, so there are more of them than of the whole numbers. The theory of infinite sets begins.
Richard Dedekind, Georg Cantor
› Sources (2)
- Dedekind, R. (1872). Stetigkeit und irrationale Zahlen. Vieweg, Braunschweig.
- Cantor, G. (1874). Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Journal für die reine und angewandte Mathematik 77: 258–262.
1902REFORMULATION
Lebesgue's integral
Henri Lebesgue's thesis measures the size of general sets and builds an integral on that measure. Riemann slices the domain; Lebesgue slices the range. His integral handles functions like Dirichlet's, and it behaves well under limits, which Riemann's does not. Measure theory became the language of modern analysis and, through Kolmogorov, of probability.
Henri Lebesgue
› Sources (1)
- Lebesgue, H. (1902). Intégrale, longueur, aire. Annali di Matematica Pura ed Applicata 7: 231–359.

Open problems

Where the map runs out

No open problems are recorded here. This field's unanswered questions moved into its successors: Continuous Optimisation + Ergodic Theory + Probability Theory + Set Theory + Stochastic Processes.