## Program

### Updated A.Y. 2020-2021

**Program**

**Set theory**. Quantifiers, subsets. Operations among sets (union, intersection, subtraction, complement) and their properties.

**Numerical sets**. Integer and rational numbers. Minimum and maximum of a set. Lower and upper bounds, inferior and superior of a set. Unbounded sets. Incompleteness of the set of rational numbers. Construction of the set of real numbers. Irrational numbers.

**Topology**. Distance between points, neighborhoods, limit points, interior points, open and closed sets.

**Functions**. Domain and image. Injective, surjective and bijective functions. Increasing and decreasing functions. Composition and inverse. Power functions and their inverse functions. Necessary and sufficient condition for invertibility. The exponential function as limit of sequences of rational numbers. Logarithms and their properties. Construction of the graph of the inverse function.

**Sequences**. Formal definition of limit of a sequence. Necessary conditions for the existence of the limit. Examples of sequences for which the limit does not exist. Absolute value theorem. Notable limits and techniques to compute limits of sequences. Indeterminate forms. Comparison theorem and applications (limit of n*sin(1/n), limit of a^n). Limits of powers, exponentials, factorials and their combinations. Theorem on increasing/decreasing sequences. The Euler sequence and the Euler number. Geometric sums and geometric series.

**Limits of functions**. Formal definition of limit of functions at a point and at infinite. Left and right limits. Necessary and sufficient conditions for the existence of the limit in terms of left and right limits. Vertical and horizontal asymptotes. Notable limits and techniques to compute limits of functions. Limits of powers, exponentials and logarithms. Indeterminate forms.

**Continuity**. Formal definition of continuity. Conditions for continuity. Non-continuous functions and their classification. Weierstrass theorem, intermediate zero theorem and applications.

**Differentiability and applications**. Incremental ratio and formal definition of differentiability. Geometrical interpetation of the derivative. Non-differentiable functions and their classifications. Proof that differentiability implies continuity. Derivative of elementary functions (powers, exponentials, logarithms, trigonometric functions). Differentiation rules. Derivative of composite function and theorem on differentiability of inverse function (derivative of arcsine, arccos, arctan). Monotonicity of differentiable functions. Local maxima, local minima and inflection points with horizontal tangent, Fermat's theorem and critical points. Convexity, concavity and higher order derivatives. The second derivative test for identifying local maxima and local minima. Sufficient conditions for identifying inflection points. De L'Hopital rule. Sketching the graph of a function.

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**Objectives**

Knowing and understanding the basic principles of mathematical analysis, starting from the foundations of topology. Applying those principles to economic modeling with a rigorous approach.

**Teaching method**

Lectures (ITA: Lezioni frontali) and tutorials (ITA: esercitazioni)

**Taching material**

The notes of the course

**Further readings**

Knut Sydsæter and Peter Hammond

*Essential mathematics for economic analysis.*