The real and complex numbers.
Review of elementary set theory and of functions between sets. The set of real numbers R and its characterisation as an ordered field with the existence of supremum property. Existence of n-th roots of positive real numbers. The extended real line. Euclidean spaces. The complex field C. Algebraic and trigonometric forms. De Moivre's formula, n-th roots. The fundamental theorem of algebra. Sets of the same cardinality. Finite and infinite sets. Countable sets. Sets of the continuum cardinality. Uncountability of R.
Metric spaces and limits of sequences.
Definition of metric space, examples and metric balls. Bounded, open, closed, compact sets. The extended real number system as a metric space. Sequences. Convergent sequences in metric spaces and their properties. Cauchy sequences and complete metric spaces. Subsequences. Sequences in R. Limits and operations on limits. Monotone sequences. Definition of the Nepero's number e and applications.
Series of real numbers.
Series in R. Convergent, divergent and irregular series. Cauchy's criterion for convergence. Series with positive terms and convergence criteria: comparison, ratio and root, condensation. Absolute convergence of a series. Alternating sign series and Leibniz's criterion for convergence.
Limits and continuity.
Limits of functions. Equivalent definition using sequences. Continuity of functions between metric spaces. Preimages of open sets. Relationship between continuity and compactness. Continuity of the composition of functions. Uniform continuity. Real functions of one real variable. Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities. Weierstrass Theorem. Intermediate value theorem and applications. Continuity of the inverse function.
Differential calculus for real functions of one real variable.
Differentiability and the definition of derivative. Derivatives of elementary functions. Computation of derivatives: algebraic operations, composition of functions and inverse function. The theorems of Rolle, Lagrange, Cauchy. Higher order derivatives.
Applications of differential calculus to the study of functions: monotonicity, local and global optimisation. De L'Hospital Theorem. Taylor formulas and applications.
The final program will we published at the end of classes on the Ariel website https://mtaralloam1.ariel.ctu.unimi.it