There are two long outstanding constructivity problems concerning
real-valued functions on the closed unit interval:
(i) Is every function continuous?
(ii) Is every continuous function uniformly continuous?
In this paper the method of reliefs is introduced to aid the study of
these problems. While so far leading to no definitive solutions, the
method has yielded a few partial results, and might be used as a basis for
further investigations.
A relief is a family of sets which is used to generate a real-valued
function as in the classical proof of Urysohn's Lemma. The numerical
content of Urysohn's method is determined here and the results are applied
to the continuity problems.
The main results are the following. For the first continuity problem,
a certain type of counterexample is given which tends to indicate that,
even in the classically true special case of monotone functions which
approximate intermediate values, it is not constructively true that every
function is continuous. For the second problem, a constructive proof of
uniform continuity is given in the special case of monotone continuous
The methods used are in accord with the principles of Bishop's
Foundations of Constructive Analysis, 1967.
1980 Mathematics Subject classification.
Primary 26A15; Secondary 03F65, 26A48, 54C30
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