Real analysis
The rigorous development of calculus: the real numbers built from completeness, and limits, continuity, differentiation, integration and convergence proved rather than computed. The corpus holds two accounts, trench-real-analysis and baggett-analysis-single-variable.
The usual description of the subject — the one Trench himself gives — is that it fills the gaps left in the development of calculus. A calculus course gets the results and defers the proofs; analysis is where the deferral is paid off, starting from the completeness axiom that the earlier course assumed without saying so.
The arc
- The real numbers — completeness, suprema, the structure the calculus sequence took on trust. 2–4. One variable — differential calculus, integral calculus, infinite sequences and series. 5–7. Several variables — real-valued functions, vector-valued functions, multiple integrals.
- Metric spaces — the abstraction the previous seven chapters were instances of.
A second account that draws the subject's boundary elsewhere
baggett-analysis-single-variable covers the same one-variable material and then declines to stop where Trench stops. Its argument is historical: real and complex analysis were not developed separately, so teaching them separately is "unnecessary and potentially confusing." What it gives up instead is dimension — several variables are "a very different kettle of fish" — so where Trench runs one variable → several variables → metric spaces, Baggett runs one variable → the complex plane.
The two therefore disagree about what the next course is, not about any theorem. Three differences worth keeping:
- Where the reals come from. Both build on completeness. Trench takes the completeness axiom in chapter 1; Baggett takes the same properties in chapter 1 and puts the construction by Dedekind cuts, plus uniqueness of a complete ordered field up to isomorphism, in an appendix he says a complete mathematics education ought to include.
- What counts as integrable. Trench develops the Riemann integral. Baggett defines the integrable functions as uniform limits of step functions, states outright that this is a smaller class, and takes the loss deliberately to make uniform convergence carry more of the book.
- Where Green's Theorem goes. In Trench it lives in the several-variable chapters, as it does in corral-vector-calculus and keisler-elementary-calculus (see vector-calculus). In Baggett it is proved in the plane and then spent immediately, as the thing Cauchy's Theorem is deduced from.
Where it sits in the foundational fork
This spoke's live tension (synthesis → the foundational fork) is whether the ε-δ definition earned the curriculum on merit or held it by inertia after Robinson's 1960 result removed the reason for it. Analysis is the course where that choice is made explicit for the first time, so it is the natural place to look for an argument.
Trench does not give one. Chapter 1 builds the reals and the ε-δ development follows, with no mention of the alternative — the same silence Corral keeps, from a more rigorous author.
The corpus therefore now holds two texts that assume the standard foundation and one that argues against it, and still nothing that argues for it. Whether the silence is inertia or settled professional judgement is exactly what the spoke cannot decide from these sources, and the count moving from one to two does not decide it either — it only makes the absence more conspicuous.
Third assumer, 2026-08-13, and the silence is now loud. baggett-analysis-single-variable does not use the word infinitesimal anywhere in 229 pages — in a book that spends an appendix constructing ℝ from Dedekind cuts and proving a complete ordered field unique up to isomorphism. Trench's silence was informative because analysis is where the choice becomes explicit; Baggett's is more so, because he is the one author here who goes all the way down to the construction and still does not say why this construction rather than Robinson's. Written in 2006, forty-six years after the hyperreals were made rigorous.
It is still not evidence against Keisler. It is a third data point for the inertia reading and none for the merit reading, from the source best placed to have supplied one.
Related
bishop-review-keisler · trench-real-analysis · baggett-analysis-single-variable · william-trench · lawrence-baggett · infinitesimal-calculus · keisler-elementary-calculus · complex-analysis · vector-calculus · mathematics