#Contents# * table of contents {:toc} ## Definition ## ### In Archimedean ordered fields ### Let $F$ be an [[Archimedean ordered field]] and let $$F_{+} \coloneqq \sum_{a:F} 0 \lt a$$ be the positive elements in $F$. A function $f:F \to F$ is __pointwise continuous__ in $F$ if $$isPointwiseContinuous(f) \coloneqq \prod_{x:F} \prod_{\epsilon:F_{+}} \prod_{y:F} \Vert \sum_{\delta:F_{+}} (\vert x - y \vert \lt \delta) \to (\vert f(x) - f(y) \vert \lt \epsilon) \Vert$$ ## See also ## * [[Archimedean ordered field]] * [[differentiable function]] * [[uniformly continuous function]]