Problems Based Course in Advanced Calculus

John M. Erdman, Portland State University, OR.

• IntervalsToplogy of the real lineContinuous functions from $\mathbb{R}$ to $\mathbb{R}$Sequences of real numbersConnectedness and the intermediate value theorem;Compactness and the extreme value theorem;Limits of real valued functions;Differentiation of real valued functions;Metric spaces;Interiors, closures, and boundaries;The topology of metric spaces;Sequences in metric spaces;Uniform convergence;More on continuity and limits;Compact metric spaces;Sequenctial characterization of compactness;Connectedness;Complete spaces;A fixed point theorem;Vector spaces;Linearity;Norms;Continuity and linearity;The Cauchy integral;Differential calculus;Partial derivatives and iterated integrals;Computations in $\mathbb{R}^n$;Infinite series;The implicit function theorem;Higher order derivatives;Quantifiers;Sets;Special subsets of $\mathbb{R}$;Logical connectives;Writing mathematics;Set operations;Arithmetic;Order propertiers of $\mathbb{R}$;Natural numbers and mathematical induction;Least upper bounds and greatest lower bounds;Products, relations, and functions;Properties of functions;Functions that have inverses;Products;Finite and infinite sets;Countable and uncountable sets;Bibliography;Index.