We introduce Z-stability, a notion capturing the intuition that if a function f maps a metric space into a normed space and if the norm of f(x) is small, then x is close to a zero of f. Working in Bishop's constructive setting, we first study pointwise versions of Z-stability and the related notion of good behaviour for functions. We then present a recursive counterexample to the classical argument for passing from pointwise Z-stability to a uniform version on compact metric spaces. In order to effect this passage constructively, we bring into play the positivity principle, equivalent to Brouwer's fan theorem for detachable bars, and the limited anti-Specker property, an intuitionistic counterpart to sequential compactness. The final section deals with connections between the limited anti-Specker property, positivity properties, and (potentially) Brouwer's fan theorem for detachable bars.

Abstract This paper provides a Bishop-style constructive analysis of the contrapositive of the statement that a continuous homomorphism ofRonto a compact abelian group is periodic. It is shown that, subject to a weak locatedness hypothesis, ifGis a complete (metric) abelian group that is the range of a continuous isomorphism fromR, thenGis noncompact. A special case occurs whenGsatisfies a certain local path-connectedness condition at 0. A number of results about one-one and injective mappings are proved en route to the main theorem. A Brouwerian example shows that some of our results are the best possible in a constructive framework.

Abstract We prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into ℝ is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.

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.

Abstract In the constructive theory of uniform spaces there occurs a technique of proof in which the application of a weak form of the law of excluded middle is circumvented by purely analytic means. The essence of this proof–technique is extracted and then applied in several different situations.

Abstract. It is shown constructively that a strongly extensional function of bounded variation on an interval is regulated, in a sequential sense that is classically equivalent to the usual one.

A mapping u: X → Y between metric spaces is sequentially continuous if for each sequence (xn) converging to x ∈ X, (u(xn)) converges to u(x). It is well known in classical mathematics that a sequentially continuous mapping between metric spaces is continuous; but, as all proofs of this result involve the law of excluded middle, there appears to be a constructive distinction between sequential continuity and continuity. Although this distinction is worth exploring in its own right, there is another reason why sequential continuity is interesting to the constructive mathematician: Ishihara [8] has a version of Banach's inverse mapping theorem in functional analysis that involves the sequential continuity, rather than continuity, of the linear mappings; if this result could be upgraded by deleting the word “sequential”, then we could prove constructively the standard versions of the inverse mapping theorem and the closed graph theorem. Troelstra [9] showed that in Brouwer's intuitionistic mathematics (INT) a sequentially continuous mapping on a separable metric space is continuous. On the other hand, Ishihara [6, 7] proved constructively that the continuity of sequentially continuous mappings on a separable metric space is equivalent to a certain boundedness principle for subsets of ℕ; in the same paper, he showed that the latter principle holds within the recursive constructive mathematics (RUSS) of the Markov School. Since it is not known whether that principle holds within Bishop's constructive mathematics (BISH), of which INT and RUSS are models and which can be regarded as the constructive core of mathematics, the exploration of sequential continuity within BISH holds some interest.