Slides & board photos, material covered

Lecture notes: chapters 1-23 (WARNING: These can change day to day.)

Board photos & synopsis of material covered:

lecture 1, March 30: Quick intro to the course. Review of metric space theory and associated topology notions. Continuity of functions in metric spaces. Sum, Product and Quotient Rules for real-valued functions. Composition Rule. Continuity of polynomials and rational functions. Sequential characterization of continuity: continuous functions map convergent sequences to convergent sequences. Topological characterization of continuity: preimages of open sets are open. (slides)
lecture 2, April 1: Open and closed maps. Bijective open maps have continuous inverse. A continuous image of a compact set is compact. A continuous function achieves its minimum and maximum on a compact set. Cauchy continuity and uniform continuity. (photos)
lecture 3, April 3: Two theorems on Cauchy/uniform continuity: 1) A continuous function on a compact set is uniformly (and thus Cauchy) continuous. 2) A densely defined uniformly (or Cauchy) continuous functions taking values in a complete metric space admits a unique continuous extension to the closure of the domain. (photos)
lecture 4, April 6: The Intermediate Value Theorem for continuous functions on bounded closed intervals in ℝ. Applications: Existence of real roots for odd-degree polynomials, existence of fixed points for continuous function mapping a closed interval to itself (i.e., a one-dimensional version of Brower's fixed point theorem), etc. The notion of connectedness in topological spaces. The topological IVT: A continuous image of a connected space is connected. (photos)
discussion, April 7: Some facts from topology: Disjoint compact sets have positive distance from each other. Every uniformly continuous function admits a modulus of continuity. Preimage of a topology is the coarsest topology that makes the function continuous. Path connectedness implies connectedness but not the other way round. (photos)
lecture 5, April 8: A limit of a function between two metric spaces. Uniqueness of the limit. Relation to continuity. Sequential characterization of a limit. Limit from a set. Right/left limits for a function on ℝ Existence of a limit equivalent to right and left limit existing and equal to each other. Right and left continuity. (photos)
lecture 6, April 10: Removable discontinuity. Discontinuities of the first and second kind. A function with a discontinuity of the second kind at all rationals (and continuous at the irrationals). The set of discontinuity points is a ℱ_σ-set. The set of rationals is not a 𝒢_δ-set and, consequently, there is no function whose discontinuity points are the irrationals. (photos)
lecture 7, April 13: The derivative of a real-valued function of a real variable. Left and right differentiability. Linear approximation. Computing the derivative of powers. Sum, product and chain rule. (photos)
discussion, April 14: Some facts from topology: Compactness, separablility, connectedness and pathwise connectedness are preserved under continuous maps. An open set is an ℱ_σ-set in metric spaces, but not in topological spaces in general. Power rule for derivative. Quotient rule for derivative. A construction of a function whose set of discontinuities is a given closed set. (photos)
lecture 8, April 15: The inverse-function rule for the derivative. Applications to differentiating fractional powers. The first-derivative test for local extrema. The Mean-Value Theorems of Rolle, Lagrange and Cauchy. Application to monotone function and differential inequality. Darboux' Theorem showing that the derivative, if defined everywhere, has the intermediate-value property. (photos)
lecture 9, April 17: Further applications of MVTs: l'Hôspital's Rule proved using Cauchy's MVT. Higher derivatives. Relation of the second derivative to convexity, the Second-derivative test. Taylor's theorem and its proof. (photos)
lecture 10, April 20: Multivariate functions of multiple variables. Definition of differentiability, partial derivative and directional derivative. Differentiability implies the other two; the total derivative is given as the Jacobian matrix. Counterexamples to the converse. Existence of partial derivatives in a neighborhood and their continuity at the point implies differentiability at the point. The concept of the matrix norm. (photos)
discussion, April 21: The total variation of a function is left/right continuous at the points where the function is left/right continuous. The exponentials are convex; proof via the AMGM inequality extended to more general arithmetic and geometric means. Differentiability of exponentials. The Euler number. (photos)
lecture 11, April 22: The matrix norm is a norm. A comparison with the Euclidean norm. Explicating the error in linear approximation. Sum rule, product rule and quotient rule of multivariate differentiation. Inverse matrix and its perturbation via matrix norm. (photos)
lecture 12, April 27: The inverse-function rule. The statement of the Inverse function theorem. Uniform matrix-norm bound on the derivative implies Lipschitz property. Proof of the two key steps of the Inverse function theorem, to be wrapped up next time. (photos)
discussion, April 28: Solving the midterm problems: l'Hospital's rule used once not twice in 2nd derivative problem, the total variation problem, the right continuity of right-limit function, fixed point without Banach contraction problem. Newton's method analyzed using the MVT. (photos)
lecture 13, April 29: Wrapping up the proof of the Inverse function theorem. Remarks on needed conditions and extensions. The key idea: invertibility of the linear approximation. Same idea used to motivate the setting of the Implicit function theorem. The proof by reduction to the Inverse function theorem. (photos)
lecture 14, May 1: Mutlivariate first derivative test. Higher multivariate derivatives. Equality of mixed partials via Clairaut/Schwarz' theorem. The Hessian and its positive semidefiniteness. The second derivative test. (photos)
lecture 15, May 4: The Riemann integral as the limit of the Riemann sum over marked partitions whose mesh tends to zero. The Cauchy criterion for Riemann integrability. The Riemann integrability implies boundedness. The linearity and monotonicity of the integral. The additivity under subdivision of integration domain. (photos)
discussion, May 5: Implicit differentiation problems: slope of a curve given in parametric form, two variable defined via three equations with five unknowns. Local extrema/saddle points of a function of two variables, role of and criteria for positive semi-definiteness. (photos)
lecture 16, May 6: The Darboux integral via upper and lower Darboux sums. Key criterion for Darboux integrability: existence of a partition bringing the upper and lower Darboux sums within ε of each other. Equivalence of Riemann integrability and Darboux integrability. (photos)
lecture 17, May 8: Finishing the proof of equivalence. The Darboux criterion helps in proving that a continuous function of a Riemann integrable function in Riemann integrable. Same for the product of Riemann integrable functions. Regularity conditions for Riemann integrability: continuity or finite number of discontinuities. Further conditions: existence of a limit at every point, absence of discontinuities of second kind. Statement of Lebesgue/Vitali theorem characterizing Riemann integrability by zero length of the set of discontinuity points. (photos)
lecture 18, May 11: The antiderivative and its role in Newton's integration theory. Riemann integral is a Lipschitz function of its limits. Differentiating the integral with respect to its upper limit yields the integral at its continuity points. The Fundamental Theorem of Calculus I. Examples where continuity is not required. The FTC II: The integral of a Riemann-integrable derivative F' is the difference F(b)-F(a). Applications: Integration by parts, Substitution Rule, Taylor's theorem with a remainder. (photos)
lecture 19, May 12: Volterra's example of a function that is differentiable everywhere with derivative bounded but not continuous in a set of non-zero length -- and thus not Riemann integrable. Proof uses the concept of a fat Cantor set. Improper integral and examples. (photos)
lecture 20, May 13: Beyond Riemann integration theory: the basic idea of Lebesgue integral is partition the range rather than the domain of the function. The Henstock-Kurzweil (HK), a.k.a. the gauge integral. The HK integral subsumes the Riemann integral and integrates the Dirichlet function. The FTCII in HK integration: The HK integral integrates all derivatives to the difference of the function at the endpoints. (photos)
lecture 21, May 15: Stietljes integral and the concept of Riemann-Stieltjes integrability. Use in physics, differential geometry and probability. Linearity in both functions. Integrability on larger interval implies integrability on subintervals and related additivity of integrals. The requirement of no common discontinuity points. Reduction to the Riemann integral. Integration by parts and substitution rule. Sufficient conditions: continuity and bounded variation. The Young integral. (photos)
lecture 22, May 18: Problem of exchange of limit. Uniform vs pointwise convergence. Interpretation via supremum metric. Uniform convergence preserves continuity. Weierstrass M-test and application to power series. Uniform convergence preserves Riemann integrability and makes integrals converge. Uniform Cauchy property of derivatives plus convergence at the point implies uniform convergence and differentiability of the limit. (slides)