nLab pointwise continuous function

Idea

The concept of continuous functions familiar from epsilontic analysis.

Definition

Definition

(pointwise continuous function 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 $c$:

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

Definition

(pointwise continuous function between Archimedean ordered fields)

Let $F$ and $K$ be Archimedean ordered fields and let

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

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

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

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

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

Definition

(pointwise continuous function between metric spaces)

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, and let

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

be the positive real numbers. A function $f:X \to Y$ is continuous at a point $c \in X$ if

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

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

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

Definition

(pointwise continuous function between uniform spaces)

Let $(X, \mathcal{U}(X), \approx)$ and $(Y, \mathcal{U}(Y), \approx)$ be uniform spaces. A function $f:X \to Y$ is continuous at a point $c \in X$ if

$$isContinuousAt(f, c) \coloneqq \forall E \in \mathcal{U}(Y). \forall x \in X. \exists D \in \mathcal{U}(X). x \approx_D c \to f(x) \approx_E f(c))$$

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

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

Definition

(pointwise continuous function between generalised filter spaces)

Let $(X, \mathcal{F}(X), \mathrm{isLimit}_X)$ and $(Y, \mathcal{F}(Y), \mathrm{isLimit}_Y)$ be generalised filter spaces. A function $f:X \to Y$ is continuous at a point $c \in X$ if

$$isContinuousAt(f, c) \coloneqq \forall F \in \mathcal{F}(X).\mathrm{isLimit}_X(F, c) \to \mathrm{isLimit}_Y(f(F), f(c)))$$

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

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

Definition

(pointwise continuous function 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 $c \in S$:

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

As structure

In dependent type theory, one could change the universal quantifiers and existential quantifiers in the definition of uniformly continuous function into dependent product types and dependent sum types.

Definition

Let $\mathrm{R}_+ \coloneqq \sum_{x:\mathbb{R}} \epsilon \gt 0$ denote the positive real numbers. Given metric spaces $(X, d_X)$ and $(Y, d_Y)$, a pointwise continuous function between $X$ and $Y$ is a function $f:X \to Y$ between their underlying sets with a dependent function which says:

For all elements $a:X$ and for all positive real number $\epsilon \gt 0$, there is as structure a positive real number $\delta \gt 0$ such that for all elements $b:X$, $\delta_Y(f(a), f(b))$ is less than $\epsilon$ whenever $\delta_X(a, b)$ is less than $\delta$

$$\prod_{a:X} \prod_{\epsilon:\mathrm{R}_+} \sum_{\delta:\mathbb{R}_+} \prod_{b:X} (\delta_X(a, b) \lt \delta) \to (\delta_Y(f(a), f(b)) \lt \epsilon)$$

By the type theoretic axiom of choice, which is simply the distributivity of dependent function types over dependent sum types, this is the same as saying that

There exists as structure a function $\omega:X \to (\mathrm{R}_+ \to \mathrm{R}_+)$ such that for all elements $a:X$, for all positive real numbers $\epsilon \gt 0$ and for all elements $b:X$, $\delta_Y(f(a), f(b))$ is less than $\epsilon$ whenever $\delta_X(a, b)$ is less than $\omega(\epsilon)$

$$\sum_{\omega:X \to (\mathrm{R}_+ \to \mathrm{R}_+)} \prod_{a:X} \prod_{\epsilon:\mathrm{R}_+} \prod_{b:X} (\delta_X(a, b) \lt \omega(\epsilon)) \to (\delta_Y(f(a), f(b)) \lt \epsilon)$$

There exists a similar definition for uniform spaces:

Definition

Given uniform spaces $(X, \mathcal{U}(X), \approx)$ and $(Y, \mathcal{U}(Y), \approx)$, a pointwise continuous function between $X$ and $Y$ is a function $f:X \to Y$ with a dependent function which says:

For all elements $x:A$ and for all entourages $E:\mathcal{U}(Y)$, there is as structure an entourage $D:\mathcal{U}(X)$ such that for all elements $b:X$, $f(a) \approx_{E} f(b)$ whenever $a \approx_{D} b$

$$\prod_{a:X} \prod_{E:\mathcal{U}(Y)} \sum_{D:\mathcal{U}(X)} \prod_{b:X} (a \approx_{D} b) \to (f(a) \approx_{E} f(b))$$

By the type theoretic axiom of choice, which is simply the distributivity of dependent function types over dependent sum types, this is the same as saying that

There exists as structure a function $\omega:X \to (\mathcal{U}(Y) \to \mathcal{U}(X))$ between the sets of entourages such that for all elements $a:X$ and entourages $E:\mathcal{U}(Y)$ and for all elements $b:X$, $f(a) \approx_{E} f(b)$ whenever $a \approx_{\omega(E)} b$

$$\sum_{\omega:X \to (\mathcal{U}(Y) \to \mathcal{U}(X))} \prod_{a:X} \prod_{E:\mathcal{U}(Y)} \prod_{b:X} (a \approx_{\omega(a, E)} b) \to (f(a) \approx_{E} f(b))$$

See also

• limit of a function

• differentiable function

• uniformly continuous function