We study classes of continuous functions on **R**^{n}
that can be approximated in various degree by uniformly
continuous ones (uniformly approachable functions).
It was proved in [BDP1:
A. Berarducci, D. Dikranjan, and J. Pelant,
*Functions with distant fibers and uniform continuity*, preprint]
that
no polynomial function can distinguish between them.
We construct examples that distinguish these classes
(answering a question from [BDP1]) and we offer
appropriate forms of uniform approachability that enable
us to obtain a general theorem on coincidence
in the class of *all*
continuous functions.

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**Last modified June 20, 2001.**