1.4 MARK MANDELKERN 4
It is merely a strong, classically equivalent, form of inclusion between
the sets of the relief (see Theorem 6.6(iii) and paragraphs 6.7 and 6.12).
Next, conditions on the relief are determined under which the generated
function is continuous, is uniformly continuous, or possesses various
other continuity and intermediate value properties (see Sections 7 and 8).
The method of reliefs may be considered a reduction of the concept of
function to the concept of set. The fundamental sets in constructive
analysis are the located sets. In [M53 the concept of located set on the
line is reduced to the concept of number. To the extent that located sets
will suffice for reliefs (for example, see paragraph 7.4 and Theorem 7.7),
functions are thus reduced to numbers.
1.4. The main results. Two main applications of the method of
reliefs have been obtained. The first, Theorem 13.6, is a constructive
proof of a special case of the uniform continuity principle:
THEOREM. Every monotone continuous real-valued function on the closed
unit interval is uniformly continuous.
The second main result concerns the following special case of the
continuity principle:
LIMITED CONTINUITY PRINCIPLE (LCP). Every monotone real-valued
function on the closed unit interval which approximates intermediate
values is continuous.
Since LCP is classically true, whereas CP is not, it is in some
respects more appropriate to investigate the constructive content of LCP
(see further remarks in Section 16). A still more limited case of CP,
involving a stronger monotonicity condition, is constructively true; see
Theorem 13.3.
A strong indication that LCP itself is constructively invalid is given
in a counterexample which is similar to a Brouwerian counterexample, but
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