  A growing body of literature suggests that odd-frequency superconducting pair
amplitudes can be generated in normal metal-superconductor junctions. The
emergence of odd-frequency pairing in these systems is often attributed to the
breaking of translation invariance. In this work, we study the pair symmetry of
a one-dimensional $s$-wave superconductor in the presence of a single
non-magnetic impurity and demonstrate that translation symmetry breaking is not
sufficient for inducing odd-frequency pairing. We consider three kinds of
impurities: a local perturbation of the chemical potential, an impurity
possessing a quantum energy level, and a local perturbation of the
superconducting gap. Surprisingly, we find local perturbations of the chemical
potential do not induce any odd-frequency pairing, despite the fact that they
break translation invariance. Moreover, although odd-frequency can be induced
by both the quantum impurity and the perturbation of the gap, we find these
odd-frequency amplitudes emerge from entirely different kinds of scattering
processes. The quantum impurity generates odd-frequency pairs by allowing one
of the quasiparticles belonging to an equal-time Cooper pair to tunnel onto the
impurity state and then back to the superconductor, giving rise to
odd-frequency amplitudes with a temporal broadening inversely proportional to
the energy level of the impurity. In contrast to this, the perturbation of the
gap leads to odd-frequency pairing by "gluing-together" normal state
quasiparticles from different points in space and time, leading to
odd-frequency amplitudes which are very localized in the time domain.
