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
Received by the editors 11 August 1981.