Contents

# Contents

## 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 preconvergence spaces)

Let $(X, \mathcal{F}(X), \mathrm{isLimit}_X)$ and $(Y, \mathcal{F}(Y), \mathrm{isLimit}_Y)$ be preconvergence 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

Last revised on July 5, 2024 at 13:41:26. See the history of this page for a list of all contributions to it.