Real analysis

Analysis is what you get when you refuse to say “approaches” without a definition. A sequence (aₙ) converges to L if for every ε > 0 there exists an integer N such that n > N implies |aₙ − L| < ε. The quantifier order is the whole content: you must answer every tolerance, and you may use a later starting index for a tighter tolerance. A limit of a function is the same idea with x → a instead of n → ∞, and with x ≠ a so the value f(a) is irrelevant.

The rational numbers have holes. The set of rationals whose square is less than 2 is bounded above and has no least upper bound inside the rationals. The real numbers are defined, or axiomatized, so that every nonempty set bounded above has a least upper bound. From that single completeness axiom you get that monotone bounded sequences converge, that Cauchy sequences converge, and that the intermediate value theorem is true. Without it, calculus is a collection of formal tricks that sometimes fail.

f is continuous at a if for every ε > 0 there exists δ > 0 such that |x − a| < δ implies |f(x) − f(a)| < ε. Uniform continuity demands one δ that works for the whole domain, for a given ε. On a closed bounded interval every continuous function is uniformly continuous, attains its bounds, and is Riemann integrable. The mean value theorem says that between a and b there is a point where the instantaneous rate equals the average rate. It is the bridge from local information, the derivative, to global information, the function.