1. PROBLEMS, METHODS, AND MAIN RESULTS 1.1. The problems. The notions of number, of function, and of continuity have seen a torturous development. Many of the perplexities and controversies of the early eighteenth century, though seemingly resolved by the beginning of this century, are in fact still with us today in the form of constructivity problems. There are two main outstanding problems in continuity - to prove or to counterexample the following statements: CONTINUITY PRINCIPLE (CP). Every real-valued function on the closed unit interval is continuous. UNIFORM CONTINUITY PRINCIPLE (UCP). Every continuous real-valued function on the closed unit interval is uniformly continuous. The classical (that is, preconstructive) solutions to these problems are both constructively invalid. A typical classical counterexample to the continuity principle, the function with value 0 at the point 0 and value 1 elsewhere is not constructively defined on the entire interval, because we possess no finite procedure for deciding whether an arbitrary point of the interval coincides with the point 0, or lies elsewhere. The classical proofs of the uniform continuity principle are also constructively invalid. For example, the Heine-Borel Theorem, on which one proof is based, does not explicitly construct finitely many neighborhoods which cover the interval, but merely proves their "existence" nonconstructively. That is, their nonexistence is contradictory. Received by the editors 11 August 1981. 1
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