Slides & board photos, material covered

Lecture notes: (WARNING: These can change day to day.)

Board photos & synopsis of material covered:

lecture 1, April 5: Quick intro to the course. 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. Open and closed maps.
lecture 2, April 6: Topological characterization of continuity. Open and closed maps. Compactness of the image of a compact set under a continuous map. Open bijections have continuous inverse. A continuous function on a compact set achieves its min and max. Cauchy continuity and relation to continuity.
lecture 3, April 7: Uniform continuity and relation to Cauchy continuity and continuity. Uniform continuity of continuous functions on compact sets. Cauchy continuity under total boundedness. A continuous extension of uniformly/Cauchy continuous functions into complete spaces. Existence of exponential function.
lecture 4, April 10: Quantitative versions of uniform continuity: Lipschitz and Hölder continuity, the notions of oscillation and modulus of continuity. Intermediate Value Theorem of Bolzano (1817). Applications: Odd-degree polynomials have roots, continuous embeddings of intervals have fixed points. Brower's fixed point theorem. The Borsuk-Ulam Theorem. Topological IVT via connectivity. Path connectivity vs connectivity.
lecture 5, April 12: Limit of a function, uniqueness. Continuity implies limit, limit implies existence of continuous modification (eliminating a removable singularity). Sequential characterization. Limes superior and inferior, characterization of limit by equality of these. Limit on a set, left and right limits for functions on ℝ. Discontinuities of first and second kind. Monotone functions have no discontinuities of second kind.
lecture 6, April 14: Monotone functions have only countably many discontinuities. Functions of bounded variation. Jordan decomposition via total variation. Uniqueness via positive and negative variations. Rectifiability of curves.
lecture 7, April 17: Derivative of a real valued function of one real variable. Differentiability implies continuity. Connection with linear approximation. Sum and product rule, chain rule and inverse function rule. First-derivative test. Mean-Value Theorems of Rolle, Lagrange and Cauchy.
lecture 8, April 19: Applications of Mean-Value Theorems: Monotone differentiable functions have derivative of one sign. Derivative of a differentiable function has no discontinuities of first kind (but those of second kind may occur densely). l'Hospital's Rule and its proof from Cauchy's MVT. Taylor's theorem via Mean Value Theorem (Rolle suffices)
lecture 9, April 21: Riemann integral: motivation, definitions of marked partition, mesh of partition and Riemann sum. Notion of a function being Riemann integrable. Linearity of integral. Riemann integrable functions are necessarily bounded. Additivity of integral under partition of the domain.
lecture 10, April 24: Darboux' approach to Riemann integral: upper and lower Darboux sums and their ordering for any pair of partitions. Upper and lower Darboux integral and associate notion of integrability. Riemann integrability implies Darboux integrability.
lecture 11, April 26: Equivalence between Riemann and Darboux integrability for bounded functions on bounded intervals. Playing with criterion of Riemann integrability arising from equivalence with Darboux integrabiliy. Riemann integrability of f and g implies Riemann integrability of |f| and f.g. Riemann integrability of continuous functions via uniform continuity. Reducing regularity: functions with finitely many discontinuities, those with no discontinuities of second kind or even indicator function of the ternary Cantor set.
lecture 12, April 28: Lebesgue's characterization of Riemann integrability by the set of discontinuity points having zero length. A proof (avoiding measure theory) with some details relegated to the notes.
lecture 13, May 1: Newton's integral and Fundamental Theorem of Calculus (FTC). FTC in Riemann's integration theory: The derivative of integral exists whenever integrand is continuous or has a limit. Integral of Riemann-integrable derivative of a function is the difference of the function at the limits of integration. Integration by parts and Substitution Rule.
lecture 14, May 3: Taylor's Theorem with remainder; proof from FTC and Integration by Parts via induction. Stieltjes integral: definition, notations. Motivations from probability and economics. Linearity and additivity. Necessary condition of no common discontinuity points. No need for boundedness of f on intervals of constancy of g for f to be integrable w.r.t. g. Generalized Stieltjes integrability.
lecture 15, May 8: Stieltjes integral continued: Relation to the Riemann integral. Integration by parts and substitution. Sufficient conditions for Stieltjes integrability: continuity of f with bounded variation of g, the Loeve-Young inequality and the Young integral.
lecture 16, May 10: Wrapping up integration. Deficiencies of Riemann integral. Behavior under pointwise limits: Osgood's Bounded Convergence Theorem. Lebesgue integral, measurability. The main defect (f integrable requires |f| integrable). Henstock-Kurzweil integral: definition and basic facts (see the notes for details).
lecture 17, May 12: Uniform convergence. Problem of exchange of limits. Pointwise and uniform convergence of functions. Continuity is preserved under uniform convergence. Exchange of limits (of functions) under uniform convergence. Power series are continuous within radius of convergence. Preservation of Riemann integrability and convergence of Riemann integrals under uniform convergence.
lecture 18, May 15: Uniformly Cauchy sequences of functions. Equivalence with uniformly convergent sequences for functions taking values in a complete metric space. Uniform convergence of derivatives plus convergence at one point implies uniform convergence of both the functions and their derivatives. Application to term-by-term differentiation of power series on their interval of (absolute) convergence. Definition and properties of (natural) exponential, sine and cosine function defined using their Taylor series. Definition of Euler number e and Ludolph number π (after Ludolph van Ceulen who over the course of his life in mid 17th century calculated π to 35 decimal points).
lecture 19, May 17: A bit more on (natural) exponential, sine and cosine functions. Irrationality of Euler's number. Conditions for uniform convergence: Dini's theorem stating that a monotone sequence of continuous function converging to a continuous functions on a compact set converges uniformly. Weierstrass M-test. Construction of a function that is differentiable with f' bounded but discontinuous at all rationals. Necessary conditions for uniform continuity: boundedness and equicontinuity.
lecture 20, May 19: Equicontinuity at a point and equicontinuity. Necessity of equicontinuity for uniform convergence of continuous functions. Pointwise convergent, equicontinuous sequences of functions have continuous limits. Equicontinuity and pointwise convergence on a dense set imply pointwise convergence everywhere. Theorem of Arzelà and Ascoli: Equicontinuous, pointwise-convergent sequence of functions on a compact set converges uniformly. Uniform equicontinuity. Equicontinuity on compact sets implies uniform equicontinuity. Another version of above Arzelà-Ascoli theorem: Uniformly equicontinuous, pointwise-convergence sequence of function on a totally bounded space converges uniformly.
lecture 21, May 22: The Arzelà-Ascoli Theorem: An equicontinuous sequence of functions between compact spaces contains a uniformly convergent subsequence. Application to finding a candidate of a minimizer of an integral functional. The space C(X,Y) and C_b(X,Y) with supremum metric. Completeness of C_b(X,Y) for Y complete. A rewrite of the Bolzano-Weierstrass and the Arzelà-Ascoli Theorem in this context. Functional version of Arzelà-Ascoli Theorem: A subset of the space of continuous functions between compact spaces is compact if and only if its closed, bounded and equicontinuous.
lecture 22, May 24: The Weierstrass approximation theorem and proof via Bernstein polynomials and facts about Binomial distribution in probability. (No probability required.) Corollaries: C([0,1],ℝ) is separable. A continuous functions orthogonal (via integral) to all monomials vanishes. The notion of an algebra separating points and not-vanishing at a point. The statement of the Stone-Weierstrass Theorem.
lecture 23, May 31: The proof of the Stone-Weierstrass Theorem for real valued functions. Linear combination of product functions are dense in continuous functions on product spaces. Density of trigonometric functions in periodic functions. Complex valued Stone-Weierstrass Theorem.
lecture 24, June 2: Analytic functions. Definition and connection with Taylor series. Infinite differentiability on interval of convergence. Example of analytic functions. Analyticity of the logarithm. Sum, product and composition of of analytic functions in analytic. Inverse function analyticity. Connection with holomorphic functions in the complex plane.
lecture 25, June 5: Trigonometric/Fourier series. Motivation via Fourier's solution of initial/boundary-value problem for the one-dimensional heat equation. Extracting the coefficients via integrals. Extension to trigonometric and complex-exponential series. Fourier coefficients. Absolute summability of Fourier coefficients of f implies uniform convergence of Fourier series to f. (Proof next time.)
lecture 26, June 7: Proof that a Fourier polynomials of f with summable Fourier coefficients converges to f uniformly: Integral representation via Dirichlet and Fejér kernels. Fejér's theorem: Fourier series of a continuous function converges in Ceàsaro sense. Regularity (twice continuous differentiability) implies uniform convergence. L²-inner product and L²-norm on the space of periodic continuous functions.