  We show that topology can protect exponentially localized, zero energy edge
modes at critical points between one-dimensional symmetry protected topological
phases. This is possible even without gapped degrees of freedom in the bulk
---in contrast to recent work on edge modes in gapless chains. We present an
intuitive picture for the existence of these edge modes in the case of
non-interacting spinless fermions with time reversal symmetry (BDI class of the
tenfold way). The stability of this phenomenon relies on a topological
invariant defined in terms of a complex function, counting its zeros and poles
inside the unit circle. This invariant can prevent two models described by the
\emph{same} conformal field theory (CFT) from being smoothly connected. A full
classification of critical phases in the non-interacting BDI class is obtained:
each phase is labeled by the central charge of the CFT, $c \in
\frac{1}{2}\mathbb N$, and the topological invariant, $\omega \in \mathbb Z$.
Moreover, $c$ is determined by the difference in the number of edge modes
between the phases neighboring the transition. Numerical simulations show that
the topological edge modes of critical chains can be stable in the presence of
interactions and disorder.
