Unit 1- Real functions of a single real variable

Sets N,Z,Q, R. Real numbers. Infimum and supremum, maximum and minimum of subsets of R. Extended real numbers R*.  Real functions of a single real variable. Main features, domain and range. Infimum and supremum, absolute maximum and minimum of a function. Injective, surjective, bijective functions. Compound and inverse function. Elementary functions. Deductible graphs. 

Unit 2- Limits

Topology of  R. Definition of a limit. Uniqueness of limits theorem. Permanence principle.Existence of limits theorem:limit comparison test, existence of limit theorem for monotone functions. Continuous functions. Discontinuities.  Continuous functions on closed bounded sets Weierstrass's Theorem. Bolzano's Theorem.  Darboux's Theorem. Limits. Indeterminate forms. notable special limits and applications. Infinite and infinitesimal and their comparisons. Landau symbol: o (o piccolo), ~ (asintotico). Asyntotics. 

Unit 3- Derivatives

Derivative of a function. Geometrical meaning of the derivative and tangent line equation. Non-derivability points. Derivability and continuity. Elementary functions derivatives. Derivation rule. higher order derivates. De l'Hôpital Theorem. Compound and inverse function derivative. Sufficient condition for derivability. Fermat Theorem. Rolle Theorem. Lagrange Theorem and corollaries.  Taylor andMc Laurin formulas and applications. Convexity and concavity of a function and inflection points. 

Unit 4- Study of a function and real functions in two variables

Study of a function. Real functions in two real variables: domain, level and partial derivatives.