  It is often thought that emergent phenomena in topological phases of matter
are destroyed when tuning to a critical point. In particular, topologically
protected edge states supposedly delocalize when the bulk correlation length
diverges. We show that this is not true in general. Edge states of topological
insulators or superconductors remain exponentially localized---despite a
vanishing band gap---if the transition increases the topological index. This
applies to all classes where the topological classification is larger than
$Z_2$, notably including Chern insulators. Moreover, these edge states are
stable to disorder, unlike in topological semi-metals. This new phenomenon is
explained by generalizing band (or mass) inversion---a unifying perspective on
topological insulators---to kinetic inversion. In the spirit of the
bulk-boundary correspondence, we also identify topological invariants at
criticality, which take half-integer values and separate topologically-distinct
universality classes by a multi-critical point. This work enlarges the scope of
topological protection and stability by showing that bulk energy gaps can be
unnecessary. Experimental probes and stability to interactions are discussed.
