Abstract In the framework of Bishop’s constructive mathematics we introduce co-convexity as a property of subsets B of ${\left\{ {0,1} \right\}^{\rm{*}}}$ , the set of finite binary sequences, and prove that co-convex bars are uniform. Moreover, we establish a canonical correspondence between detachable subsets B of ${\left\{ {0,1} \right\}^{\rm{*}}}$ and uniformly continuous functions f defined on the unit interval such that B is a bar if and only if the corresponding function f is positive-valued, B is a uniform bar if and only if f has positive infimum, and B is co-convex if and only if f satisfies a weak convexity condition.

We consider a special case of Dickson's lemma: for any two functions $f,g$ on the natural numbers there are two numbers $i

Abstract We prove constructively that the weak König lemma and quantifier-free number–number choice imply that every pointwise continuous function from Cantor space into Baire space has a modulus of uniform continuity.

Abstract The existence and uniqueness of a maximum point for a continuous real–valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.