Existential statements seem to admit a constructive proof without
countable choice only if the object to be constructed is uniquely determined,
or is intended as an approximate solution of the problem in question.
This conjecture is substantiated by examining some basic tools of
mathematical analysis from a choice-free constructive point of view,
concentrating on Dedekind cuts as an appropriate notion of real numbers.
As a complement, the question whether densely defined continuous functions
do approximate intermediate values is reduced to connectivity properties
of the corresponding domains.
