nLab pointwise continuous function

Idea

The concept of continuous functions familiar from epsilontic analysis.

Definition

In the real numbers

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

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

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

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

$f$ is pointwise continuous in $\mathbb{R}$ if it is continuous at all points $c$:

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

In Archimedean fields

Let $F$ be an Archimedean field and let

$$F_{+} \coloneqq \{a \in F \vert 0 \lt a\}$$

be the positive elements in $F$. A function $f:F \to F$ is continuous at a point $c \in F$ if

$$isContinuousAt(f, c) \coloneqq \forall \epsilon \in F_{+}. \forall x \in F. \exists \delta \in F_{+}. (\max(x - c, c - x) \lt \delta) \to (\max(f(x) - f(c), f(c) - f(x)) \lt \epsilon)$$

$f$ is pointwise continuous in $F$ if it is continuous at all points $c$:

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

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 $X$ (the “underlying set”);

2. 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.

Example

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

$$x \sim_\epsilon y \coloneqq {\Vert x-y\Vert \lt \epsilon} \,.$$

Definition

(epsilontic definition of pointwise continuity)

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

$$f \;\colon\; X \longrightarrow Y$$

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

$$x \sim_\delta^X y \;\Rightarrow\; f(x) \sim_\epsilon^Y f(y)$$

The function $f$ is called pointwise continuous if it is continuous at every point $x \in X$.

In preconvergence spaces

Let $S$ and $T$ be preconvergence spaces. A function $f:S \to T$ is continuous at a point $c \in S$ if

$$isContinuousAt(f, c) \coloneqq \forall I \in DirectedSet_\mathcal{U}. \forall x \in I \to S. isLimit_S(x, c) \to isLimit_T(f \circ x, f(c))$$

$f$ is pointwise continuous if it is continuous at all points $c$:

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

In function limit spaces

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 $f(c)$.

$$\lim_{x \to c} f(x) = f(c)$$

$f$ is pointwise continuous on $S$ if it is continuous at all points $c \in S$:

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

See also

• limit of a function
• differentiable function
• uniformly continuous function