  The first part of the paper is to improve the fundamental theory of
isoparametric functions on general Riemannian manifolds. Next we focus our
attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and
the Gromoll-Meyer sphere. In particular, as one of main results we prove: there
exists no properly transnormal function on any exotic 4-sphere if it exists.
Furthermore, by projecting an $S^3$-invariant isoparametric function on
$Sp(2)$, we construct a properly transnormal but not an isoparametric function
on the Gromoll-Meyer sphere with two points as the focal varieties.
