  We study the global boundedness of the solutions of a non-smooth forced
oscillator with a periodic and real analytic forcing. We show that the impact
map associated with this discontinuous equation becomes a real analytic and
exact symplectic map when written in suitable canonical coordinates. By an
accurate study of the behaviour of the map for large amplitudes and by
employing a parametrization KAM theorem, we show that the periodic solutions of
the unperturbed oscillator persist as two-dimensional tori under conditions
that depend on the Diophantine conditions of the frequency, but are independent
on both the amplitude of the orbit and of the specific value of the frequency.
This allows the construction of a sequence of nested invariant tori of
increasing amplitude that confine the solutions within them, ensuring their
boundedness. The same construction may be useful to address such persistence
problem for a larger class of non-smooth forced oscillators.
