  It is quite well-known from Kurt Godel's (1931) ground-breaking result on the
Incompleteness Theorem that rudimentary relations (i.e., those definable by
bounded formulae) are primitive recursive, and that primitive recursive
functions are representable in sufficiently strong arithmetical theories. It is
also known, though perhaps not as well-known as the former one, that some
primitive recursive relations are not rudimentary. We present a simple and
elementary proof of this fact in the first part of the paper. In the second
part, we review some possible notions of representability of functions studied
in the literature, and give a new proof of the equivalence of the weak
representability with the (strong) representability of functions in
sufficiently strong arithmetical theories. Our results shed some new light on
the notions of rudimentary, primitive recursive, and representable functions
and relations, and clarify, hopefully, some misunderstandings and confusing
errors in the literature.
