[[!redirects pointwise continuous function]] [[!redirects uniformly continuous function]] #Contents# * table of contents {:toc} ## Definition ## ### In rational numbers ### Let $\mathbb{Q}$ be the [[rational numbers]], and let $I$ be a [[closed interval]] in $\mathbb{Q}$. An function $f:I \to \mathbb{Q}$ is __continuous at a point__ $c:I$ $$isContinuousAt(f, c) \coloneqq \prod_{\epsilon:\mathbb{Q}_{+}} \prod_{x:I} \Vert \sum_{\delta:\mathbb{Q}_{+}} (\vert x - c \vert \lt \delta) \to (\vert f(x) - f(c) \vert \lt \epsilon) \Vert$$ $f$ is __pointwise continuous__ in $I$ if it is continuous at all points $c$: $$isPointwiseContinuous(f) \coloneqq \prod_{c:I} isContinuousAt(f, c)$$ $f$ is __uniformly continuous__ in $I$ if $$isUniformlyContinuous(f) \coloneqq \prod_{\epsilon:\mathbb{Q}_{+}} \Vert \sum_{\delta:\mathbb{Q}_{+}} \prod_{x:I} \prod_{y:I} (x \sim_\delta y) \to (f(x) \sim_\epsilon f(y)) \Vert$$ ### 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__ $c:S$ if the [[limit of a function|limit]] of $f$ approaching $c$ exists and is equal to $f(c)$ $$isContinuousAt(f, c) \coloneqq isContr(\lim_{x \to c} f(x) = f(c))$$ $f$ 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 $$isUniformlyContinuous(f) \coloneqq \prod_{\epsilon:V} \Vert \sum_{\delta:T} \prod_{x:S} \prod_{y:S} (x \sim_\delta y) \to (f(x) \sim_\epsilon f(y)) \Vert$$ ## See also ## * [[premetric space]] * [[limit of a function]]