nLab pointwise continuous function

Contents

Contents

Idea

The concept of continuous functions familiar from epsilontic analysis.

Definition

In the real numbers

Let \mathbb{R} be the real numbers and let

+{a|0<a}\mathbb{R}_{+} \coloneqq \{a \in \mathbb{R} \vert 0 \lt a\}

be the positive elements in \mathbb{R}. A function f:f:\mathbb{R} \to \mathbb{R} is continuous at a point cc \in \mathbb{R} if

isContinuousAt(f,c)ϵ +.δ +.x.(|xc|<δ)(|f(x)f(c)|<ϵ)isContinuousAt(f, c) \coloneqq \forall \epsilon \in \mathbb{R}_{+}. \exists \delta \in \mathbb{R}_{+}. \forall x \in \mathbb{R}. (\vert x - c \vert \lt \delta) \to (\vert f(x) - f(c) \vert \lt \epsilon)

ff is pointwise continuous in \mathbb{R} if it is continuous at all points cc:

isPointwiseContinuous(f)c.isContinuousAt(f,c)isPointwiseContinuous(f) \coloneqq \forall c \in \mathbb{R}. isContinuousAt(f, c)

In Archimedean fields

Let FF be an Archimedean field and let

F +{aF|0<a}F_{+} \coloneqq \{a \in F \vert 0 \lt a\}

be the positive elements in FF. A function f:FFf:F \to F is continuous at a point cFc \in F if

isContinuousAt(f,c)ϵF +.xF.δF +.(max(xc,cx)<δ)(max(f(x)f(c),f(c)f(x))<ϵ)isContinuousAt(f, c) \coloneqq \forall \epsilon \in F_{+}. \forall x \in F. \exists \delta \in F_{+}. (\max(x - c, c - x) \lt \delta) \to (\max(f(x) - f(c), f(c) - f(x)) \lt \epsilon)

ff is pointwise continuous in FF if it is continuous at all points cc:

isPointwiseContinuous(f)cF.isContinuousAt(f,c)isPointwiseContinuous(f) \coloneqq \forall c \in F. isContinuousAt(f, c)

In premetric spaces

We state the definition of pointwise continuity in terms of epsilontic analysis, definition below.

Definition

A premetric space is

  1. a set XX (the “underlying set”);

  2. a ternary relation () ()():X× +×XΩ(-)\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.

Example

Every normed vector space (V,)(V, {\Vert -\Vert}) becomes a premetric space according to def. by setting

x ϵyxy<ϵ. x \sim_\epsilon y \coloneqq {\Vert x-y\Vert \lt \epsilon} \,.
Definition

(epsilontic definition of pointwise continuity)

For (X, X)(X,\sim^X) and (Y, Y)(Y,\sim^Y) two premetric spaces (def. ), then a function

f:XY f \;\colon\; X \longrightarrow Y

is said to be continuous at a point xXx \in X if for every ϵ\epsilon there exists δ\delta such that

x δ Xyf(x) ϵ Yf(y) x \sim_\delta^X y \;\Rightarrow\; f(x) \sim_\epsilon^Y f(y)

The function ff is called pointwise continuous if it is continuous at every point xXx \in X.

In preconvergence spaces

Let SS and TT be preconvergence spaces. A function f:STf:S \to T is continuous at a point cSc \in S if

isContinuousAt(f,c)IDirectedSet 𝒰.xIS.isLimit S(x,c)isLimit T(fx,f(c))isContinuousAt(f, c) \coloneqq \forall I \in DirectedSet_\mathcal{U}. \forall x \in I \to S. isLimit_S(x, c) \to isLimit_T(f \circ x, f(c))

ff is pointwise continuous if it is continuous at all points cc:

isPointwiseContinuous(f)cS.isContinuousAt(f,c)isPointwiseContinuous(f) \coloneqq \forall c \in S. isContinuousAt(f, c)

In function limit spaces

Let TT be a Hausdorff function limit space, and let STS \subseteq T be a subset of TT. A function f:STf:S \to T is continuous at a point cSc \in S if the limit of f f approaching c c is equal to f(c)f(c).

lim xcf(x)=f(c)\lim_{x \to c} f(x) = f(c)

ff is pointwise continuous on SS if it is continuous at all points cSc \in S:

isPointwiseContinuous(f)cS.lim xcf(x)=f(c)isPointwiseContinuous(f) \coloneqq \forall c \in S. \lim_{x \to c} f(x) = f(c)

See also

Last revised on October 19, 2022 at 19:37:29. See the history of this page for a list of all contributions to it.