Optimization. A first course of mathematics for economists Xavier Martinez-Giralt Universitat Aut`onoma de Barcelona [email protected]

I.2.- Continuity

OPT – p.1/13

Continuity Intuition A function f is continuous when given any two arbitrarily close points of its domain generate images arbitrarily close. Formal definition - preliminaries Let A ⊂ IRn , f : A → IRm . Let x0 be an accumulation point of A. We say that b ∈ IRm is the limit of f at the point x0 ∈ A, limx→x0 f (x) = b, if given an arbitrary ε > 0, ∃δ > 0 (dependent of f, x0 and ε) such that ∀x ∈ A, x 6= x0 , kx − x0 k < δ implies kf (x) − bk < ε. Remark 1: If x0 is not accumulation point, 6 ∃x 6= x0 , x ∈ A close to x0 , and the definition is empty of content. Remark 2: It may happen that the limit of a function at a point does not exist. But whenever it exists, it is unique. OPT – p.2/13

Continuity (2) Formal definitions Let A ⊂ IRn , f : A → IRm . Let x0 ∈ A. We say that f is continuous at a point x0 ∈ A if ∀ε > 0, ∃δ > 0 such that ∀x ∈ A, kx − x0 k < δ implies kf (x) − f (x0 )k < ε. We say that f is continuous on B ⊂ A if it is continuous ∀x ∈ B . When we say that “f is continuous" it means that f is continuous on its domain A. Continuity of f in [a, b]: f continuous in (a, b), i.e. limx→x0 f (x) = f (x0 ) and f right-continuous at a i.e. limx→a+ f (x) = f (a) and f left-continuous at b i.e. limx→b− f (x) = f (b)

OPT – p.3/13

Continuity - Illustration f (x)

f (x)

f

f (x0 ) + ε ε ε f (x0 ) − ε

ε x 0

ε

δ

δ

x a

x0

b

f is continuous at x0

f is not continuous at 0

|x − x0 | < δ ⇒ |f (x) − f (x0 )| < ε

OPT – p.4/13

Algebra with Continuous Functions Preliminaries Let f : A → IRm and g : B → IRp be two functions such that f (A) ⊂ B . The composition of function g with function f , denoted as g ◦ f : A → Rp is defined as x 7→ g(f (x)). Let f : A → IRm and g : B → IRp be two continuous functions such that f (A) ⊂ B . Then, g ◦ f : A → Rp is a continuous function. Let A ⊂ IRn . Let x0 be an accumulation point of A. Let f : A → Rm and g : A → Rm be two functions. Assume limx→x0 f (x) = a and limx→x0 g(x) = b. Then, limx→x0 (f + g)(x) = a + b, where f + g : A → Rm is defined as (f + g)(x) = f (x) + g(x). Then, limx→x0 (f · g)(x) = ab, where f · g : A → Rm is defined as (f · g)(x) = f (x)g(x).

OPT – p.5/13

Algebra with Continuous Functions (2) Preliminaries (cont’d) Assume limx→x0 f (x) = a 6= 0 and f is not zero in a neighborhood of x0 Then, limx→x0 (g/f )(x) = b/a, where g/f : A → Rm is defined as (g/f )(x) = g(x)/f (x). Algebra with Continuous Functions Let A ⊂ IRn . Let x0 be an accumulation point of A. Let f : A → Rm and g : A → Rm be two continuous functions at x0 . Then, f + g : A → Rm is continuous at x0 . Let f : A → Rm and g : A → Rm be two continuous functions at x0 . Then, f · g : A → Rm is continuous at x0 . Let f : A → Rm and g : A → Rm be two continuous functions at x0 . Let f (x0 ) 6= 0. Then, f is not zero in a neighborhood U of x0 , and g/f : U → Rm is continuous at x0 . OPT – p.6/13

Boundedness Theorem Intuition A continuous function defined on a compact set attains its maximum and minimum values at some point of the set. Remark 1: A continuous function need not be bounded. Example: f (x) = 1/x, x ∈ (0, 1). Remark 2: A continuous and bounded function need not reach its maximum value at any point of its domain. Example: f (x) = x, x ∈ [0, 1). f (x)

f (x)

1/x 0

(

)

x 1

0[

)

x 1

OPT – p.7/13

Boundedness Theorem (2) Theorem Let A ⊂ IRn and f : A → IR be continuous. Let K ⊂ A be a compact set. Then, f is bounded on K , that is B = {f (x)|x ∈ K} is a bounded set. Furthermore, ∃(x0 , x1 ) ∈ K such that f (x0 ) = inf(B) and f (x1 ) = sup(B) where sup(B) denotes the absolute maximum of f on K and inf(B) denotes the absolute minimum of f on K .

OPT – p.8/13

Intermediate Value Theorem The Intermediate Value Theorem Let A ⊂ IRn Let f : A → IR be a continuous function. Let K ⊂ A be a connected set. Consider a, b ∈ K . Then, ∀c ∈ [f (a), f (b)], ∃z ∈ K such that f (z) = c. Bolzano’s Theorem Let f : A → IR be a continuous function. Let K ⊂ A be a connected set. Consider a, b ∈ K , such that signf (a) 6= signf (b). There ∃z ∈ K such that f (z) = 0.

OPT – p.9/13

Intermediate Value and Bolzano Theorems - Illustration f (x)

f (x) f

f (b)

f (x)

f (b)

f

c f c c f (a)

f (a) a

z

b

a

x

b

z

x

x

f (x) f f (b) a

z

b

x

f(a)

OPT – p.10/13

Weaker concepts of continuity directional continuity right continuity - no jump when appraoching the limit from the right left continuity - no jump when appraoching the limit from the left semi-continuity upper semi-continuity: jumps if any, only go up lower semi-continuity: jumps if any, only go down

OPT – p.11/13

Weaker concepts of continuity- Illustration

f (x) f (c)

ε ε f (c) f (x)

ε

δ c

ε

x c+δ

Right continuity ∀x ∈ (c, c + δ), |(f (x) − f (c)| < ε

δ c−δ x

c

Left continuity ∀x ∈ (c − δ, c), |(f (x) − f (c)| < ε

OPT – p.12/13

Weaker concepts of continuity- Illustration

f (c) + ε ε f (x) f (c) f (x)

δ x

δ c

x

Upper semi-continuity ∀x, |x − c| < δ, f (x) ≤ f (c) + ε

f (x) f (c) f (x)

ε

f (c) − ε

δ x

δ c

x

Lower semi-continuity ∀x, |x − c| < δ, f (x) ≥ f (c) − ε

OPT – p.13/13