[[!redirects pointwise continuous function]] [[!redirects uniformly continuous function]] #Contents# * table of contents {:toc} ## Definition ## ### In rational numbers ### Let $\mathbb{Q}$ be the [[rational numbers]]. An function $f:\mathbb{Q} \to \mathbb{Q}$ is __continuous at a point__ $c:\mathbb{Q}$ $$isContinuousAt(f, c) \coloneqq \prod_{\epsilon:\mathbb{Q}_{+}} \prod_{x:\mathbb{Q}} \Vert \sum_{\delta:\mathbb{Q}_{+}} (\vert x - c \vert \lt \delta) \to (\vert f(x) - f(c) \vert \lt \epsilon) \Vert$$ $f$ is __pointwise continuous__ in $\mathbb{Q}$ if it is continuous at all points $c$: $$isPointwiseContinuous(f) \coloneqq \prod_{c:\mathbb{Q}} isContinuousAt(f, c)$$ $f$ is __uniformly continuous__ in $\mathbb{Q}$ if $$isUniformlyContinuous(f) \coloneqq \prod_{\epsilon:\mathbb{Q}_{+}} \Vert \sum_{\delta:\mathbb{Q}_{+}} \prod_{x:\mathbb{Q}} \prod_{y:\mathbb{Q}} (x \sim_\delta y) \to (f(x) \sim_\epsilon f(y)) \Vert$$ ### In premetric spaces ### Let $R$ be a dense integral subdomain of the [[rational numbers]] $\mathbb{Q}$ and let $R_{+}$ be the positive terms of $R$. Let $S$ and $T$ be $R_{+}$-[[premetric space]]s. A function $f:S \to T$ is __continuous at a point__ $c:S$ $$isContinuousAt(f, c) \coloneqq \prod_{\epsilon:R_{+}} \prod_{x:S} \Vert \sum_{\delta:R_{+}} (x \sim_\delta c) \to (f(x) \sim_\epsilon f(c)) \Vert$$ $f$ is __pointwise continuous__ if it is continuous at all points $c$: $$isPointwiseContinuous(f) \coloneqq \prod_{c:S} isContinuousAt(f, c)$$ $f$ is __uniformly continuous__ if $$isUniformlyContinuous(f) \coloneqq \prod_{\epsilon:R_{+}} \Vert \sum_{\delta:R_{+}} \prod_{x:S} \prod_{y:S} (x \sim_\delta y) \to (f(x) \sim_\epsilon f(y)) \Vert$$ ### Most general definition ### Let $S$ be a type with a [[predicate]] $\to_S$ between the type of all nets in $S$ $$\sum_{I:\mathcal{U}} isDirected(I) \times (I \to S)$$ and $S$ itself, and let $T$ be a type with a [[predicate]] $\to_T$ between the type of all nets in $T$ $$\sum_{I:\mathcal{U}} isDirected(I) \times (I \to T)$$ and $T$ itself. A function $f:S \to T$ is __continuous at a point__ $c:S$ $$isContinuousAt(f, c) \coloneqq \sum_{I:\mathcal{U}} isDirected(I) \times \prod_{x:I \to S} (x \to_S c) \to (f \circ x \to_T f(c))$$ $f$ is __pointwise continuous__ if it is continuous at all points $c$: $$isPointwiseContinuous(f) \coloneqq \prod_{c:S} isContinuousAt(f, c)$$ ## See also ## * [[premetric space]] * [[limit of a function]]