  We are concerned with the formation of singularities and the existence of
global continuous solutions of the Cauchy problem for the one-dimensional
non-isentropic Euler equations for compressible fluids. For the isentropic
Euler equations, we pinpoint a necessary and sufficient condition for the
formation of singularities of solutions with large initial data that allow a
far-field vacuum -- there exists a compression in the initial data. For the
non-isentropic Euler equations, we identify a sufficient condition for the
formation of singularities of solutions with large initial data that allow a
far-field vacuum -- there exists a strong compression in the initial data.
Furthermore, we identify two new phenomena -- decompression and de-rarefaction
-- for the non-isentropic Euler flows, different from the isentropic flows, via
constructing two respective solutions. For the decompression phenomenon, we
construct a first global continuous non-isentropic solution, even though
initial data contain a weak compression, by solving an inverse Goursat problem,
so that the solution is smooth, except on several characteristic curves across
which the solution has a weak discontinuity (i.e., only Lipschitz continuity).
For the de-rarefaction phenomenon, we construct a continuous non-isentropic
solution whose initial data contain isentropic rarefactions (i.e., without
compression) and a locally stationary varying entropy profile, for which the
solution still forms a shock wave in a finite time.
