  We investigate the metric mean dimension of subshifts of compact type. We
prove that the metric mean dimensions of a continuous map and its inverse limit
coincide, generalizing Bowen's entropy formula. Building upon this result, we
extend the notion of metric mean dimension to discontinuous maps in terms of
suitable subshifts. As an application, we show that the metric mean dimension
of the Gauss map and that of induced maps of the Manneville-Pomeau family is
equal to the box dimension of the corresponding set of discontinuity points,
which also coincides with a critical parameter of the pressure operator
associated to the geometric potential.
