Limits and Continuity

Let $f : D \subset R^{n} \to R$
For $x_{0} \in \bar{D}$ , $lim_{x \to x_{0}} f (x) = L$ if:

\begin{matrix} \forall ε > 0, \exists δ > 0 such that if x \in D and 0 < | | x - x_{0} | | < δ \\ ⟹ | f (x) - L | < δ \end{matrix}

Alternatively:
For $x_{0} \in \bar{D}$ , $lim_{x \to x_{0}} f (x) = L$ if:

\forall B_{ϵ} (L), \exists f ((B_{δ} (x_{0}) \cap D) ∖ x_{0}) \subset B_{ϵ} (L)

Say $f : D \subset R^{n} \to R^{m}$ such that:

f (\vec{x}) = [\begin{matrix} g_{1} (\vec{x}) \\ g_{2} (\vec{x}) \\ ⋮ \\ g_{m} (\vec{x}) \end{matrix}]

$lim_{\vec{x} \to \vec{x_{0}}} f (\vec{x})$ exists if and only if:

lim_{\vec{x} \to \vec{x_{0}}} g_{1} (\vec{x}), lim_{\vec{x} \to \vec{x_{0}}} g_{2} (\vec{x}), \dots, lim_{\vec{x} \to \vec{x_{0}}} g_{m} (\vec{x})

exists, where $g_{i}, i \in {1, 2, \dots, m}$ are real-valued

Alternatively:
For $x_{0} \in \bar{D}$ , $lim_{x \to x_{0}} f (x) = \vec{L}$ if:

\forall B_{ϵ} (\vec{L}), \exists f ((B_{δ} (x_{0}) \cap D) ∖ x_{0}) \subset B_{ϵ} (\vec{L})

Let $f : D \subset R^{n} \to R^{m}$ :
$f$ is continuous at $x_{0}$ if $lim_{x \to x_{0}} f (x) = f (x_{0})$

Formally $f$ is continuous if:

$\forall ϵ > 0, \exists δ > 0 such that d (x, x_{0}) < δ ⟹ d (f (x), f (x_{0}) < ϵ$

Alternatviely
$f$ is continuous iff the pre-image of every open set is open.

$f : D \subset R^{n} \to R^{m}$ is Lipschitz-continuous if for some fixed $k \in R :$

| | f (x) - f (y) | | < k | | x - y | |