Entry requirements

60 credits in Mathematics. Basic Topology recommended.

Learning outcomes

The course aims at strengthening and generalizing results that the student has already learnt from previous calculus courses. In order to pass the course the student should

• be able to give an account of various consequences of the mean value theorem;
• master the concept of convergence for numerical sequences and series and for sequences and series of functions, and in particular understand the distinction between pointwise and uniform convergence;
• be able to give an account of the concept of equicontinuity;
• be able to give an account of the contraction principle and to derive some of its consequences in several variable calculus;
• be able to formulate central theorems in the area of the course and to describe the main features of their proofs;
• be able to use the theory, methods and techniques of the course to solve mathematical problems.

Content

Numerical sequences and series: limit points, upper and lower limits, Cauchy sequences, rearrangements. Continuity: discontinuities, monotonic functions. Differentiation of single-variable functions: the mean value theorem, continuity properties of derivatives, l'Hospital's rule and Taylor series. Sequences and series of functions: Uniform convergence, equicontinuous families of functions, the Arzelà-Ascoli theorem, the Stone-Weierstrass theorem. Functions of several variables: the contraction principle, inverse and implicit function theorems, the rank theorem.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.