Regrettably, there is no time for a detailed discussion on alternative definitions of upper/lower limits. Prospective analysts should nonetheless be aware of them (Definition 2 and Definition 3 in the notes linked above). The proofs of the equivalence of all these definitions are, admittedly, technical and notationally heavy, but it draws on many important ideas we have discussed thus far: sup/inf, limits, Cauchy sequences, etc.

Theorem 5.2: differentiability implies continuity.

Theorems 5.8, 5.11: local extrema, critical points, and the first derivative test.

Theorem 5.10: the mean value theorem.

Theorem 5.15: Taylor's theorem with the Lagrange form of the remainder.

Time constraints force us to abandon a detailed discussion of Riemann's integration theory in favor of that of Lebesgue. Do not, however, discount the Riemann integral, whose construction will again be relevant in the development of stochastic calculus.

Note that Rudin discusses not just the vanilla version of the Riemann integral, but the fancier Riemann–Stieltjes integral, in which one integrates againt a ``weight''' (i.e., a measure) $\alpha = \alpha(x)$. It is instructive to work through Rudin's proofs yourself in the special case $\alpha(x) = x$ (so that you are working with the usual Riemann integral) to really understand what is going on.

Highlights from this chapter include the following.

Definitions 6.1, 6.3: upper/lower Darboux sums, Riemann integrability, refinements of partitions

Theorem 6.6: an equivalent, and computationally more useful, characterization of Riemann integrability.

Theorem 6.7: justifying the ``limit as mesh size goes to zero of a Riemann sum'' nonsense (what does this limit mean in $\epsilon$-$\delta$ form?) that one encounters in a calculus class.

Theorems 6.8, 6.10: functions with finitely many points of discontinuity are integrable.

Theorem 6.19: change of variables formula.

Theorems 6.20, 6.21: the fundamental theorems of calculus.

Theorem 6.22: integration by parts, (one of) the greatest trick(s) in mathematics.

In light of Theorem 6.10, one might be tempted to investigate the necessary and sufficient condition(s) for a function to be Riemann integrable. This condition turns out to be that "the set of discontinuities has Lebesgue measure zero," which offers a compelling transition to measure theory in the second half of this semester.

We do not have time to study uniform convergence, which is the contents of [Ru, Chapter 7], in any detail. Highlights from this chapter include the following.

Definitions 7.1, 7.7: pointwise vs. uniform convergence of a seuqnece of functions.

Theorem 7.10: the Weierstraß $M$-test.

Theorem 7.12: "the uniform limit of a sequence of continuous functions is again continuous."

Definition 7.14, Theorem 7.15: "endowed with the supremum norm, the space of continuous functions on a compact set is a complete metric space."

Theorem 7.16: "the uniform limit of a sequence of Riemann integrable functions is again Riemann integrable."

Theorem 7.17: uniform convergence and differentiability (note that the statement is rather subtle).

Other classical results include Arzelà–Ascoli (Theorem 7.25), Weierstraß approximation (Theorem7.26), and Stone–Weierstraß (Theorems 7.32, 7.33).

I do not present the details of the proof in class because it is extremely technical. Some of the more subtle ideas, such as product of measures and completion of a measure, are made more evident when one moves to the more general setting of abstract measure theory [SS, Chapter 6].