Limits
It is time to discuss the notion of limits in the several variable setting, on which later subjects of mathematical analysis will be based on. The are many similarities between single and several variable limits.
Definition 1.0 (limit of a function) Let A be an open subset of ℝn, f: A → ℝ, a function of n variables, and x0 a point in A or its boundary. The limit of f(x) as x → x0 is the number ℓ if for every ε > 0 there is a δ > 0 such that for all x ∈ A with 0 < ||x − x0|| < δ we have |f(x) − ℓ| < ε.
Corollary 1.0 (permanence of the sign) Let Ω ⊆ ℝn, f: Ω → ℝ and x0 ∈ ℝn an accumulation point in Ω. If there exists limx → x0 f(x) = ℓ ∈ R, ℓ ≠ 0, there there exists a neighborhood U ⊆ ℝn of x0 such that f on U ∩ (Ω \ {x0} ) has the same sign of ℓ.
Proof - By the ε-δ definition of limit, by setting ε = ℓ/2
ℓ - |ℓ| / 2 < f(x,y) < ℓ + |ℓ| / 2
If ℓ > 0 from the first inequality
0 < ℓ/2 < f(x,y)
If ℓ < 0 from the second inequality
f(x,y) < ℓ - ℓ/2 = ℓ/2 < 0. □
Definition 4.3 (Continuity) - A function f: ℝn → ℝ is continuous at a point x0 if:
