18 Limits and Continuity
You’ve computed limits since first-year calculus — plug in, factor, cancel, apply L’Hôpital’s rule when things get stuck at \(\frac{0}{0}\). This chapter isn’t about any of that. It’s about what a limit structurally is, and it closes a gap the map flagged in its very first pages: calculus is roughly two centuries older than the theory that makes it rigorous.
18.1 The problem that took 150 years to fix
Newton and Leibniz built calculus (1660s–80s) on infinitesimals — quantities smaller than every positive number, yet not zero. The intuition worked; the foundation didn’t hold up to scrutiny. Bishop George Berkeley’s 1734 critique, calling these quantities “ghosts of departed quantities,” was a fair objection: nobody could say precisely what an infinitesimal was, only that calculus kept producing correct answers when you used them. For roughly 150 years, calculus was a spectacularly successful computational practice standing on a foundation nobody could fully defend.
Augustin-Louis Cauchy (1820s) and Karl Weierstrass (1870s) replaced infinitesimals with something that needed no mysterious infinitely-small numbers at all: the \(\varepsilon\)–\(\delta\) definition. \(\lim_{x \to a} f(x) = L\) means: for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that \(0 < |x-a| < \delta\) implies \(|f(x)-L| < \varepsilon\). No infinitesimals — just ordinary real numbers, made as small as needed, with an explicit accounting of how small “needed” has to be. This is the rigorous foundation now called real analysis, and the field’s whole reason for existing is closing exactly the gap this section opened with.
18.2 This is a definition you’ve already built
Here’s the payoff of Metric Spaces and Topological Spaces: the \(\varepsilon\)–\(\delta\) definition above is not new machinery. It is the general metric-space notion of convergence — \(y_n \to y\) when \(d(y_n, y) \to 0\) — specialized to \(\mathbb{R}\) with its ordinary distance \(d(x,y) = |x-y|\). Continuity at a point, \(\lim_{x\to a} f(x) = f(a)\), is exactly Topological Spaces’s “preimage of every open set is open,” worked out concretely for open intervals on \(\mathbb{R}\). Nothing in this chapter is conceptually new; it’s the general theory from two chapters ago, now doing the specific job it was generalized from in the first place.
18.3 The infinitesimals weren’t wrong, just early
One more twist worth knowing, briefly: in the 1960s, Abraham Robinson showed that infinitesimals can be made fully rigorous after all, using a different foundation (nonstandard models of the real numbers) than Cauchy and Weierstrass’s. Newton and Leibniz’s original intuition wasn’t mathematically unsound — it just needed roughly 300 years, and a foundation nobody in the 17th century had access to, to be justified on its own terms. This site doesn’t develop nonstandard analysis further; it’s mentioned here as one more instance of the map’s historical/formal split cutting in an unexpected direction.
18.4 What follows
Two consequences worth naming, both leaning directly on completeness — Number Systems already flagged this dependency before either theorem existed on this site. The intermediate value theorem: a continuous function on \([a,b]\) hits every value between \(f(a)\) and \(f(b)\). The extreme value theorem: a continuous function on a closed interval \([a,b]\) attains an actual maximum and minimum. Both fail over \(\mathbb{Q}\) for the same reason \(\mathbb{Q}\) failed to contain \(\sqrt{2}\): completeness is doing the work, not continuity alone.
Discontinuities come in different flavors worth having names for, even without full development here: removable (the limit exists but doesn’t match \(f(a)\), or \(f(a)\) is undefined), jump (the limit from the left and right both exist but disagree), and essential (no limit exists at all, however you approach the point).
18.5 If you’ve computed limits and used continuity
You already know how to evaluate \(\lim_{x\to 2} \frac{x^2-4}{x-2}\) by factoring, and you already reach for continuity implicitly every time you evaluate a function “by just plugging in.” What you now have that you may not have had explicitly is the actual content behind “plugging in works because \(f\) is continuous there”: it’s a promise, provable from the \(\varepsilon\)–\(\delta\) definition, that \(f\) near \(a\) behaves the way \(f(a)\) suggests it should — a promise that fails, precisely, at the discontinuities named above.
18.6 Where this leads
Differentiation, next, is a specific limit — the limit of a difference quotient — built directly on everything in this chapter.