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 one account, trench-real-analysis.
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.
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.
Related
trench-real-analysis · william-trench · infinitesimal-calculus · keisler-elementary-calculus · vector-calculus · mathematics