Functions of n real variable 1

What is a real-valued function of n real variables?

It is a function f: A ⊆ ℝⁿ → ℝ. Its input is a vector x = (x₁, x₂, ..., xₙ) ∈ A, and its output f(x) is a real number.

What are the domain, codomain, and image of f: A ⊆ ℝⁿ → ℝ?

The domain is A ⊆ ℝⁿ; the codomain is ℝ; and the image is Im(f) = {f(x) : x ∈ A} ⊆ ℝ.

What is the natural domain of a function?

The natural domain is the largest set on which the function is defined. For a real-valued function of n real variables, it is a subset of ℝⁿ and is imposed by the function's formula.

What is the graph of a function f: A → B?

The graph of f is gr(f) = {(x, f(x)) : x ∈ A} ⊆ A × B.

Where does the graph of f: A ⊆ ℝⁿ → ℝ lie?

It lies in ℝⁿ⁺¹. In particular, if f: A ⊆ ℝ² → ℝ, then gr(f) = {(x, y, z) : z = f(x, y)} ⊆ ℝ³ and is called a surface.

What is the k-level set of a scalar function f: A → ℝ?

For k ∈ ℝ, the k-level set is {f = k} = f⁻¹({k}) = {x ∈ A : f(x) = k}.

What is the k-level curve of f: A ⊆ ℝ² → ℝ?

It is the set {f = k} = {(x, y) ∈ A : f(x, y) = k}.

When is f: A ⊆ ℝⁿ → ℝ bounded from above?

f is bounded from above if there exists M ∈ ℝ such that f(x) ≤ M for every x ∈ A.

When is f: A ⊆ ℝⁿ → ℝ bounded from below?

f is bounded from below if there exists m ∈ ℝ such that f(x) ≥ m for every x ∈ A.

When is f: A ⊆ ℝⁿ → ℝ bounded?

f is bounded if its image Im(f) is bounded in ℝ; equivalently, f is bounded both from above and from below.

State the proposition characterizing a bounded function using absolute value.

A function f: A ⊆ ℝⁿ → ℝ is bounded if and only if there exists k > 0 such that |f(x)| ≤ k for every x ∈ A.

How is the supremum of f: A ⊆ ℝⁿ → ℝ defined?

sup{x ∈ A} f(x) = sup Im(f). It is the least upper bound of the image of f.

How is the infimum of f: A ⊆ ℝⁿ → ℝ defined?

inf{x ∈ A} f(x) = inf Im(f). It is the greatest lower bound of the image of f.

What is a global maximizer of f on C?

Let f: A ⊆ ℝⁿ → ℝ and C ⊆ A. A point x̂ ∈ C is a global maximizer of f on C if f(x̂) ≥ f(x) for every x ∈ C.

What is the global maximum value of f on C?

If x̂ is a global maximizer, then f(x̂) is the global maximum value of f on C, denoted max{x ∈ C} f(x).

What is arg max of f on C?

arg max{x ∈ C} f(x) = {x̂ ∈ C : f(x̂) = max{x ∈ C} f(x)}. It is the set of all global maximizers.

What is a global minimizer of f on C?

A point x̂ ∈ C is a global minimizer of f on C if f(x̂) ≤ f(x) for every x ∈ C.

What are the global minimum value and arg min of f on C?

If x̂ is a global minimizer, f(x̂) is the global minimum value, denoted min{x ∈ C} f(x). Moreover, arg min{x ∈ C} f(x) = {x̂ ∈ C : f(x̂) = min{x ∈ C} f(x)}.

What is a strong maximizer of f on C?

A point x̂ ∈ C is a strong maximizer if f(x̂) > f(x) for every x ∈ C distinct from x̂.

What is a strong minimizer of f on C?

A point x̂ ∈ C is a strong minimizer if f(x̂) < f(x) for every x ∈ C distinct from x̂.

State the theorem relating strong and unique global optimizers.

A global maximizer is strong if and only if it is unique. Analogously, a global minimizer is strong if and only if it is unique.

What is a local maximizer of f on C?

Let f: A ⊆ ℝⁿ → ℝ and C ⊆ A. A point x̂ ∈ C is a local maximizer if there exists ε > 0 such that f(x̂) ≥ f(x) for every x ∈ Bε(x̂) ∩ C.

What is a local minimizer of f on C?

A point x̂ ∈ C is a local minimizer if there exists ε > 0 such that f(x̂) ≤ f(x) for every x ∈ Bε(x̂) ∩ C.

Why does the definition of a local optimizer use Bε(x̂) ∩ C?

The intersection restricts the comparison to feasible points in C. This is especially important when x̂ lies on the boundary of C.

State the theorem characterizing when two functions represent the same ordering.

Let g, h: A ⊆ ℝⁿ → ℝ. Then g(x) ≥ g(y) if and only if h(x) ≥ h(y) for all x, y ∈ A exactly when there exists a strictly increasing function φ: Im(g) → ℝ such that h = φ ∘ g.