Real Valued Functions of Several Variables

Real Valued Functions of Several Variables: 
If D ⊂ Rⁿ the function
                     f(x) = f(x₁ , x₂ , . . . , x_n)
where  (x) ∈ Rⁿ and f : D → Rⁿ is called real valued function of several variables.
Limit: A function f (x), x ∈ D ⊂ Rⁿ has a limit l, 
i.e., lim f(x) = l, if for given ε > 0, there exists δ > 0 such that
|f(x) - l| < ε for every a ∈ D, ||x - a|| < δ.
Continuity: A function f : D → Rⁿ is continuous at x = a, if for each ε > 0, there exists δ > 0 such that
                            |f(x) - f(a)| < ε 
whenever,          ||x - a|| < δ.
and                      x ∈ D ⊂ Rⁿ
or f : D → Rⁿ is continuous at a iff  
             lim f(x) = f(a).
Uniform Continuity: A function f : D → Rⁿ is uniformly continuous on D if it continuous at every x ∈ D ⊂ Rⁿ

Some important Results:

• The range of a function continuouse on a compact set is compact.
• A real valued function continuous on a compact set is bounded and attains its bound.
• A real valued function continuous on a closed rectangle [a, b] is bounded and attains its bound.
• A function continuous on a compact domain is uniformly continuous.

Extended Real Numbers:
In mathematics, the extended real number system is obtained from the real number system R by adding two elements: + ∞ and - ∞ (read as positive infinity and negative infinity respectively). These new elements are not real numbers. it is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The extended real number system is denoted R̅ or [+ ∞ , - ∞].

The extended real number system is the set of real numbers ℝ, together with two symbols + ∞ and - ∞ which satisfied.
         (a) If x∈ℝ, then
                      x + (+ ∞) = + ∞, x + (- ∞) = -∞
                      x - (+ ∞) = - ∞, x - (- ∞) = + ∞
                      x/+ ∞ = x/- ∞ = 0
         (b) If x > 0, then,
                     x (+ ∞) = + ∞, x (- ∞) = - ∞
         (c) If x < 0, then
                     x (+ ∞) = - ∞, x (- ∞) = + ∞
         (d) (+ ∞) + (+ ∞) = (+ ∞)(+ ∞)
                                       = (- ∞)(- ∞) = + ∞
               (- ∞) + (- ∞) = (+ ∞)(- ∞) = - ∞
         (e) x ∈ℝ, then ∞ < x < + ∞.

Important: ∞ - ∞ is undefined
     0/0, ∞/∞, -∞/-∞ is also undefined
    and 0.∞ = 0.

Boundedness and extended real number system:

1. The set S ⊂ ℝ is bounded above if
                        sup S < ∞
2. The set S ⊂ ℝ is bounded below if 
                        inf S > - ∞

Integers, Rational numbers as Sub-set of real number ℝ:

Archimedes axiom: Given any real number x, there is an integer n such that x < n.

1. Every ordered field contains the integers, the natural numbers and the rational numbers.
2. Between any two real numbers is a rational number, i.e., if x < y, there is a rational number r : x < r < y.
3. The set ℤ⁺ of positive integers 1, 2, . . . is unbounded above.