  Let $\mathcal{S} = \{ \tau_n \}_{n=1}^\infty \subset (0,T)$ be an arbitrary
countable (dense) set. We show that for any given initial density and momentum,
the compressible Euler system admits (infinitely many) admissible weak
solutions that are not strongly continuous at each $\tau_n$, $n=1,2,\dots$. The
proof is based on a refined version of the oscillatory lemma of De Lellis and
Sz\' ekelyhidi with coefficients that may be discontinuous on a set of zero
Lebesgue measure.
