analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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The concept of continuous functions familiar from epsilontic analysis.
Let $\mathbb{R}$ be the real numbers and let
be the positive elements in $\mathbb{R}$. A function $f:\mathbb{R} \to \mathbb{R}$ is continuous at a point $c \in \mathbb{R}$ if
$f$ is pointwise continuous in $\mathbb{R}$ if it is continuous at all points $c$:
Let $F$ be an Archimedean field and let
be the positive elements in $F$. A function $f:F \to F$ is continuous at a point $c \in F$ if
$f$ is pointwise continuous in $F$ if it is continuous at all points $c$:
We state the definition of pointwise continuity in terms of epsilontic analysis, definition below.
A premetric space is
a set $X$ (the “underlying set”);
a ternary relation $(-)\sim_{(-)} (-)\colon X \times \mathbb{R}_+ \times X \to \Omega$ (the “premetric”) from the Cartesian product of the set with the positive real numbers and with itself again to the set of truth values.
Every normed vector space $(V, {\Vert -\Vert})$ becomes a premetric space according to def. by setting
(epsilontic definition of pointwise continuity)
For $(X,\sim^X)$ and $(Y,\sim^Y)$ two premetric spaces (def. ), then a function
is said to be continuous at a point $x \in X$ if for every $\epsilon$ there exists $\delta$ such that
The function $f$ is called pointwise continuous if it is continuous at every point $x \in X$.
Let $S$ and $T$ be preconvergence spaces. A function $f:S \to T$ is continuous at a point $c \in S$ if
$f$ is pointwise continuous if it is continuous at all points $c$:
Let $T$ be a Hausdorff function limit space, and let $S \subseteq T$ be a subset of $T$. A function $f:S \to T$ is continuous at a point $c \in S$ if the limit of $f$ approaching $c$ is equal to $f(c)$.
$f$ is pointwise continuous on $S$ if it is continuous at all points $c \in S$:
Last revised on October 19, 2022 at 19:37:29. See the history of this page for a list of all contributions to it.