ABSTRACT

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

functions.

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