789 B
789 B
Limits and continuity
Limit
Definition: let D \subseteq \mathbb{R}^m
and let f: D \to \mathbb{R}^n
, with m,n \in \mathbb{N}
. Let \mathbf{a}
be the point \mathbf{x}
approaches, then f
approaches the limit L \in \mathbb{R}^n
\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = L \iff \forall \varepsilon_{>0} \exists \delta_{>0} \Big[0 < |\mathbf{x} - \mathbf{a}|< \delta \implies |f(\mathbf{x}) - L| < \varepsilon \Big],
with \mathbf{a}, \mathbf{x} \in \mathbb{R}^m
.
Continuity
Definition: let D \subseteq \mathbb{R}^m
and let f: D \to \mathbb{R}^n
, with m,n \in \mathbb{N}
. Then f
is called continuous at \mathbf{a}
if
\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = f(\mathbf{a}),
with \mathbf{a}, \mathbf{x} \in \mathbb{R}^m
.