  In his seminal work from the 1950s, William Feller classified all
one-dimensional diffusions on $-\infty\leq a<b\leq \infty$ in terms of their
ability to access the boundary (Feller's test for explosions) and to enter the
interior from the boundary. Feller's technique is restricted to diffusion
processes as the corresponding differential generators allow explicit
computations and the use of Hille-Yosida theory. In the present article we
study exit and entrance from infinity for the most natural generalization, that
is, jump diffusions of the form \[
  dZ_t=\sigma(Z_{t-})\,dX_t, \] driven by stable L\'evy processes for
$\alpha\in (0,2)$. Many results have been proved for jump diffusions, employing
a variety of techniques developed after Feller's work but exit and entrance
from infinite boundaries has long remained open. We show that the presence of
jumps implies features not seen in the diffusive setting without drift. Finite
time explosion is possible for $\alpha\in (0,1)$, whereas entrance from
different kinds of infinity is possible for $\alpha\in [1,2)$. We derive
necessary and sufficient conditions on $\sigma$ so that (i) non-exploding
solutions exist and (ii) the corresponding transition semigroup extends to an
entrance point at `infinity'.
  Our proofs are based on very recent developments for path transformations of
stable processes via the Lamperti-Kiu representation and new Wiener-Hopf
factorisations for L\'evy processes that lie therein. The arguments draw
together original and intricate applications of results using the
Riesz-Bogdan--\.Zak transformation, entrance laws for self-similar Markov
processes, perpetual integrals of L\'evy processes and fluctuation theory,
which have not been used before in the SDE setting, thereby allowing us to
employ classical theory such as Hunt-Nagasawa duality and Getoor's
characterisation of transience and recurrence.
