Paris.

Optimization. Philippe Bich (Paris 1 Panthéon-Sorbonne and PSE) Paris, 2016.

Chapter 0: why optimization ?

Some examples: An individual optimizes his consumption basket at the supermarket (this is a behavioural model!) A social planner optimizes when takes some decisions: minimize inequalities, maximize total well being. A firm optimizes when takes some decisions: minimize inequalities, maximize total well being. Nature optimizes: brachistochrone curve. Student optimizes: maximize grade minimizing work. In finance: maximizing return, minimizing risk.

Chapter 0: why optimization ?

Some examples: An individual optimizes his consumption basket at the supermarket (this is a behavioural model!) A social planner optimizes when takes some decisions: minimize inequalities, maximize total well being. A firm optimizes when takes some decisions: minimize inequalities, maximize total well being. Nature optimizes: brachistochrone curve. Student optimizes: maximize grade minimizing work. In finance: maximizing return, minimizing risk.

Chapter 0: why optimization ?

Some examples: An individual optimizes his consumption basket at the supermarket (this is a behavioural model!) A social planner optimizes when takes some decisions: minimize inequalities, maximize total well being. A firm optimizes when takes some decisions: minimize inequalities, maximize total well being. Nature optimizes: brachistochrone curve. Student optimizes: maximize grade minimizing work. In finance: maximizing return, minimizing risk.

Chapter 0: why optimization ?

Some examples: An individual optimizes his consumption basket at the supermarket (this is a behavioural model!) A social planner optimizes when takes some decisions: minimize inequalities, maximize total well being. A firm optimizes when takes some decisions: minimize inequalities, maximize total well being. Nature optimizes: brachistochrone curve. Student optimizes: maximize grade minimizing work. In finance: maximizing return, minimizing risk.

Chapter 0: why optimization ?

Some examples: An individual optimizes his consumption basket at the supermarket (this is a behavioural model!) A social planner optimizes when takes some decisions: minimize inequalities, maximize total well being. A firm optimizes when takes some decisions: minimize inequalities, maximize total well being. Nature optimizes: brachistochrone curve. Student optimizes: maximize grade minimizing work. In finance: maximizing return, minimizing risk.

Chapter 0: why optimization ?

Some examples: An individual optimizes his consumption basket at the supermarket (this is a behavioural model!) A social planner optimizes when takes some decisions: minimize inequalities, maximize total well being. A firm optimizes when takes some decisions: minimize inequalities, maximize total well being. Nature optimizes: brachistochrone curve. Student optimizes: maximize grade minimizing work. In finance: maximizing return, minimizing risk.

Chapter 1: Reminders (real analysis) lower bound, upper bound of a subet S of R. bounded subet S of R. greatest element, least element of a subset S of R. Definition of sup S, inf S, max S, min S when S is a subset of R. Existence. Sequential characterization. Exercise: S = {(−1)n + min S ?

1 n

: n ∈ IN∗ }. What is sup S, inf S, max S,

S = {(−1)n + n : n ∈ IN∗ }. What is sup S, inf S, max S, min S ? S = { 21n + 21m : n ∈ IN∗ , m ∈ IN∗ }. What is sup S, inf S, max S, min S ?

Chapter 1: Reminders (real analysis) lower bound, upper bound of a subet S of R. bounded subet S of R. greatest element, least element of a subset S of R. Definition of sup S, inf S, max S, min S when S is a subset of R. Existence. Sequential characterization. Exercise: S = {(−1)n + min S ?

1 n

: n ∈ IN∗ }. What is sup S, inf S, max S,

S = {(−1)n + n : n ∈ IN∗ }. What is sup S, inf S, max S, min S ? S = { 21n + 21m : n ∈ IN∗ , m ∈ IN∗ }. What is sup S, inf S, max S, min S ?

Chapter 1: Reminders (real analysis) lower bound, upper bound of a subet S of R. bounded subet S of R. greatest element, least element of a subset S of R. Definition of sup S, inf S, max S, min S when S is a subset of R. Existence. Sequential characterization. Exercise: S = {(−1)n + min S ?

1 n

: n ∈ IN∗ }. What is sup S, inf S, max S,

S = {(−1)n + n : n ∈ IN∗ }. What is sup S, inf S, max S, min S ? S = { 21n + 21m : n ∈ IN∗ , m ∈ IN∗ }. What is sup S, inf S, max S, min S ?

Chapter 1: Reminders (real analysis) lower bound, upper bound of a subet S of R. bounded subet S of R. greatest element, least element of a subset S of R. Definition of sup S, inf S, max S, min S when S is a subset of R. Existence. Sequential characterization. Exercise: S = {(−1)n + min S ?

1 n

: n ∈ IN∗ }. What is sup S, inf S, max S,

S = {(−1)n + n : n ∈ IN∗ }. What is sup S, inf S, max S, min S ? S = { 21n + 21m : n ∈ IN∗ , m ∈ IN∗ }. What is sup S, inf S, max S, min S ?

Chapter 1: Reminders (real analysis) lower bound, upper bound of a subet S of R. bounded subet S of R. greatest element, least element of a subset S of R. Definition of sup S, inf S, max S, min S when S is a subset of R. Existence. Sequential characterization. Exercise: S = {(−1)n + min S ?

1 n

: n ∈ IN∗ }. What is sup S, inf S, max S,

S = {(−1)n + n : n ∈ IN∗ }. What is sup S, inf S, max S, min S ? S = { 21n + 21m : n ∈ IN∗ , m ∈ IN∗ }. What is sup S, inf S, max S, min S ?

Chapter 1: Reminders (real analysis) lower bound, upper bound of a subet S of R. bounded subet S of R. greatest element, least element of a subset S of R. Definition of sup S, inf S, max S, min S when S is a subset of R. Existence. Sequential characterization. Exercise: S = {(−1)n + min S ?

1 n

: n ∈ IN∗ }. What is sup S, inf S, max S,

S = {(−1)n + n : n ∈ IN∗ }. What is sup S, inf S, max S, min S ? S = { 21n + 21m : n ∈ IN∗ , m ∈ IN∗ }. What is sup S, inf S, max S, min S ?

Chapter 1: Reminders (real analysis) lower bound, upper bound of a subet S of R. bounded subet S of R. greatest element, least element of a subset S of R. Definition of sup S, inf S, max S, min S when S is a subset of R. Existence. Sequential characterization. Exercise: S = {(−1)n + min S ?

1 n

: n ∈ IN∗ }. What is sup S, inf S, max S,

S = {(−1)n + n : n ∈ IN∗ }. What is sup S, inf S, max S, min S ? S = { 21n + 21m : n ∈ IN∗ , m ∈ IN∗ }. What is sup S, inf S, max S, min S ?

Chapter 1: Reminders (real analysis) lower bound, upper bound of a subet S of R. bounded subet S of R. greatest element, least element of a subset S of R. Definition of sup S, inf S, max S, min S when S is a subset of R. Existence. Sequential characterization. Exercise: S = {(−1)n + min S ?

1 n

: n ∈ IN∗ }. What is sup S, inf S, max S,

S = {(−1)n + n : n ∈ IN∗ }. What is sup S, inf S, max S, min S ? S = { 21n + 21m : n ∈ IN∗ , m ∈ IN∗ }. What is sup S, inf S, max S, min S ?

Chapter 1: Reminders (real analysis)

domain of a function f : E → R. Image of a function f : E → R. lower bound, upper bound of a function f : E → R on C ⊂ E. bounded function f : E → R. supremum, infimum, maximum, minimum of f : E → R on E. Sequential Characterization of the supremum and infimum of f on E. Exercises: find infimum or minimum of f (x) = E =]0, +∞[.

1 x2

on

Chapter 1: Reminders (real analysis)

domain of a function f : E → R. Image of a function f : E → R. lower bound, upper bound of a function f : E → R on C ⊂ E. bounded function f : E → R. supremum, infimum, maximum, minimum of f : E → R on E. Sequential Characterization of the supremum and infimum of f on E. Exercises: find infimum or minimum of f (x) = E =]0, +∞[.

1 x2

on

Chapter 1: Reminders (real analysis)

domain of a function f : E → R. Image of a function f : E → R. lower bound, upper bound of a function f : E → R on C ⊂ E. bounded function f : E → R. supremum, infimum, maximum, minimum of f : E → R on E. Sequential Characterization of the supremum and infimum of f on E. Exercises: find infimum or minimum of f (x) = E =]0, +∞[.

1 x2

on

Chapter 1: Reminders (real analysis)

domain of a function f : E → R. Image of a function f : E → R. lower bound, upper bound of a function f : E → R on C ⊂ E. bounded function f : E → R. supremum, infimum, maximum, minimum of f : E → R on E. Sequential Characterization of the supremum and infimum of f on E. Exercises: find infimum or minimum of f (x) = E =]0, +∞[.

1 x2

on

Chapter 1: Reminders (real analysis)

domain of a function f : E → R. Image of a function f : E → R. lower bound, upper bound of a function f : E → R on C ⊂ E. bounded function f : E → R. supremum, infimum, maximum, minimum of f : E → R on E. Sequential Characterization of the supremum and infimum of f on E. Exercises: find infimum or minimum of f (x) = E =]0, +∞[.

1 x2

on

Chapter 1: Reminders (real analysis)

domain of a function f : E → R. Image of a function f : E → R. lower bound, upper bound of a function f : E → R on C ⊂ E. bounded function f : E → R. supremum, infimum, maximum, minimum of f : E → R on E. Sequential Characterization of the supremum and infimum of f on E. Exercises: find infimum or minimum of f (x) = E =]0, +∞[.

1 x2

on

Chapter 1: Reminders (real analysis)

domain of a function f : E → R. Image of a function f : E → R. lower bound, upper bound of a function f : E → R on C ⊂ E. bounded function f : E → R. supremum, infimum, maximum, minimum of f : E → R on E. Sequential Characterization of the supremum and infimum of f on E. Exercises: find infimum or minimum of f (x) = E =]0, +∞[.

1 x2

on

Chapter 1: Reminders (real analysis)

standard operations on IRn ; difference between vectors and points in IRn . euclidean distance, norm, scalar product in IRn . Cauchy-schwartz inequality; Triangle inequality; Closed ball, open ball.

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). Consider the maximisation and minimization problems: (P) max f (x). x∈C

(Q) min f (x). x∈C

f is the objective function, C the set of feasible points (or set of constraints).

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). Consider the maximisation and minimization problems: (P) max f (x). x∈C

(Q) min f (x). x∈C

f is the objective function, C the set of feasible points (or set of constraints).

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). Consider the maximisation and minimization problems: (P) max f (x). x∈C

(Q) min f (x). x∈C

f is the objective function, C the set of feasible points (or set of constraints).

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a solution of (P) if ... x ∈ E is a solution of (Q) has a local solution if ...

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a solution of (P) if ... x ∈ E is a solution of (Q) has a local solution if ...

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a solution of (P) if ... x ∈ E is a solution of (Q) has a local solution if ...

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a solution of (P) if ... x ∈ E is a solution of (Q) has a local solution if ...

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a solution of (P) if ... x ∈ E is a solution of (Q) has a local solution if ...

Chapter 2: Optimization (vocabulary)

Let f : E → R, and denote D(f ) the domain of f . Let C ⊂ D(f ). x ∈ E is a solution of (P) if .... x ∈ E is a solution of (Q) if ... To define local solution, we now assume D(f ) ⊂ IRn for some n. x ∈ E is a solution of (P) if ... x ∈ E is a solution of (Q) has a local solution if ...

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

A maximizing sequence for (P) is .... Let (Q) min f (x). x∈C

A minimizing sequence for (P) is .... existence

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

A maximizing sequence for (P) is .... Let (Q) min f (x). x∈C

A minimizing sequence for (P) is .... existence

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

A maximizing sequence for (P) is .... Let (Q) min f (x). x∈C

A minimizing sequence for (P) is .... existence

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

A maximizing sequence for (P) is .... Let (Q) min f (x). x∈C

A minimizing sequence for (P) is .... existence

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

A maximizing sequence for (P) is .... Let (Q) min f (x). x∈C

A minimizing sequence for (P) is .... existence

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

Aim of optimization is to compute the value of (P) (if exists), and to know if there exists one (or many) solutions of (P), and to compute the solutions. If there is no solution, the aim is to compute a maximizing sequence, if exists.

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

Aim of optimization is to compute the value of (P) (if exists), and to know if there exists one (or many) solutions of (P), and to compute the solutions. If there is no solution, the aim is to compute a maximizing sequence, if exists.

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

Aim of optimization is to compute the value of (P) (if exists), and to know if there exists one (or many) solutions of (P), and to compute the solutions. If there is no solution, the aim is to compute a maximizing sequence, if exists.

Chapter 2: Optimization (vocabulary)

Let (P) max f (x). x∈C

Aim of optimization is to compute the value of (P) (if exists), and to know if there exists one (or many) solutions of (P), and to compute the solutions. If there is no solution, the aim is to compute a maximizing sequence, if exists.

Chapter 2: Optimization (vocabulary) big ideas in optimization: Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, (see chapter 3). When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. In chapter 4, we recall important properties related to differentiability. Thus critical points of f could be candidate to be solution of an optimization problem. Under some convexity assumption, we can get the converse statement: In chapter 5, we recall important properties of convexity.

Chapter 2: Optimization (vocabulary) big ideas in optimization: Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, (see chapter 3). When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. In chapter 4, we recall important properties related to differentiability. Thus critical points of f could be candidate to be solution of an optimization problem. Under some convexity assumption, we can get the converse statement: In chapter 5, we recall important properties of convexity.

Chapter 2: Optimization (vocabulary) big ideas in optimization: Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, (see chapter 3). When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. In chapter 4, we recall important properties related to differentiability. Thus critical points of f could be candidate to be solution of an optimization problem. Under some convexity assumption, we can get the converse statement: In chapter 5, we recall important properties of convexity.

Chapter 2: Optimization (vocabulary) big ideas in optimization: Sometimes, we can be sure an optimization problem has a solution without any computation, only from "topological properties" of the problem. Typical "topological" properties are continuity (for a function), compactness (for a set), openness, closeness, (see chapter 3). When f is differentiable, it is possible, under some assumptions to be defined later, to prove that a solution of an optimization problem is a critical point of f , i.e. satisfies f 0 (x) = 0. In chapter 4, we recall important properties related to differentiability. Thus critical points of f could be candidate to be solution of an optimization problem. Under some convexity assumption, we can get the converse statement: In chapter 5, we recall important properties of convexity.

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology Section 1: Open subset, closed subset, ... We consider on Rn the euclidean distance, the associated euclidean norm and euclidean scalar product. Openess of C ⊂ Rn . Example, properties, Closeness of C ⊂ Rn . Example, properties, Sequential characterization of closeness of C. Interiors points, interior set, accumulation point, closure, boundary A subset C of Rn is said to be bounded if ... A subset C of Rn is said to be compact if ... Example: C = {(x, x1 ) : x > 0} is closed ? open ? 1 Example: C = {(x, x+1 ) : x > 0} is closed ? open ?

Chapter 3: Topology

Section 1: Open subset, closed subset, ... Exercise Consider C = {(x, y ) ∈ R2 : x 2 + y 2 < 1, x 6= 0} ∪ {(2, 1)}. open, closed ? closure ? interior ? accumulation points ? boundary ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x + y 2 ≤ 1}. is it compact ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x 4 ≤ 1}. is it compact ? Exercise Consider C = {(x, y ) ∈ R2 :| x |< 1, | y |≤ 1}. is it compact ?

Chapter 3: Topology

Section 1: Open subset, closed subset, ... Exercise Consider C = {(x, y ) ∈ R2 : x 2 + y 2 < 1, x 6= 0} ∪ {(2, 1)}. open, closed ? closure ? interior ? accumulation points ? boundary ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x + y 2 ≤ 1}. is it compact ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x 4 ≤ 1}. is it compact ? Exercise Consider C = {(x, y ) ∈ R2 :| x |< 1, | y |≤ 1}. is it compact ?

Chapter 3: Topology

Section 1: Open subset, closed subset, ... Exercise Consider C = {(x, y ) ∈ R2 : x 2 + y 2 < 1, x 6= 0} ∪ {(2, 1)}. open, closed ? closure ? interior ? accumulation points ? boundary ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x + y 2 ≤ 1}. is it compact ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x 4 ≤ 1}. is it compact ? Exercise Consider C = {(x, y ) ∈ R2 :| x |< 1, | y |≤ 1}. is it compact ?

Chapter 3: Topology

Section 1: Open subset, closed subset, ... Exercise Consider C = {(x, y ) ∈ R2 : x 2 + y 2 < 1, x 6= 0} ∪ {(2, 1)}. open, closed ? closure ? interior ? accumulation points ? boundary ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x + y 2 ≤ 1}. is it compact ? Exercise Consider C = {(x, y ) ∈ R2 : x 2 + x 4 ≤ 1}. is it compact ? Exercise Consider C = {(x, y ) ∈ R2 :| x |< 1, | y |≤ 1}. is it compact ?

Chapter 3: Topology

Section 2: Continuity Let C ⊂ R. A function f : C → R is continuous if ... Sequential characterization. link with limit. A function g : C → Rp is continuous if ... pre-image of a closed or open set by a continuous mapping f : Rn → R. Continuous mapping on a compact set. Intermediate value theorem.

Chapter 3: Topology

Section 2: Continuity Let C ⊂ R. A function f : C → R is continuous if ... Sequential characterization. link with limit. A function g : C → Rp is continuous if ... pre-image of a closed or open set by a continuous mapping f : Rn → R. Continuous mapping on a compact set. Intermediate value theorem.

Chapter 3: Topology

Section 2: Continuity Let C ⊂ R. A function f : C → R is continuous if ... Sequential characterization. link with limit. A function g : C → Rp is continuous if ... pre-image of a closed or open set by a continuous mapping f : Rn → R. Continuous mapping on a compact set. Intermediate value theorem.

Chapter 3: Topology

Section 2: Continuity Let C ⊂ R. A function f : C → R is continuous if ... Sequential characterization. link with limit. A function g : C → Rp is continuous if ... pre-image of a closed or open set by a continuous mapping f : Rn → R. Continuous mapping on a compact set. Intermediate value theorem.

Chapter 3: Topology

Section 2: Continuity Let C ⊂ R. A function f : C → R is continuous if ... Sequential characterization. link with limit. A function g : C → Rp is continuous if ... pre-image of a closed or open set by a continuous mapping f : Rn → R. Continuous mapping on a compact set. Intermediate value theorem.

Chapter 3: Topology

Section 2: Continuity C ⊂ Rn is pathwise connected if .... Bolzano theorem:

Chapter 3: Topology

Section 3: Other notions of Continuity right continuity, left continuity. upper semicontinuity; lower semicontinuity.

Chapter 3: Topology

Section 3: Other notions of Continuity right continuity, left continuity. upper semicontinuity; lower semicontinuity.

Chapter 3: Topology

Section 4: Basic geometry Hyperplane of Rn . Half-hyperplane. Representation of {x : f (x) ≤ c}. Level sets. Interpretation of maximization problem with level sets.

Chapter 3: Topology

Section 4: Basic geometry Hyperplane of Rn . Half-hyperplane. Representation of {x : f (x) ≤ c}. Level sets. Interpretation of maximization problem with level sets.

Chapter 3: Topology

Section 4: Basic geometry Hyperplane of Rn . Half-hyperplane. Representation of {x : f (x) ≤ c}. Level sets. Interpretation of maximization problem with level sets.

Chapter 3: Topology

Section 3: Other notions of Continuity right continuity, left continuity. upper semicontinuity; lower semicontinuity.

Chapter 3: Topology

Section 3: Other notions of Continuity right continuity, left continuity. upper semicontinuity; lower semicontinuity.