  Extendibility of inflationary spacetimes with flat spatial geometry is
investigated. We find that the past boundary of an inflationary spacetime
becomes a so-called parallely propagated curvature singularity if the ratio
$\dot{H}/a^2$ diverges at the boundary, where $\dot{H}$ and $a$ represent the
time derivative of the Hubble parameter and the scale factor, respectively. On
the other hand, if the ratio $\dot{H}/a^2$ converges, then the past boundary is
regular and continuously extendible. We also develop a method to judge the
continuous ($C^0$)extendibility of spacetime in the case of slow-roll inflation
driven by a canonical scalar field. As applications of this method, we find
that Starobinsky inflation has a $C^0$ parallely propagated curvature
singularity, but a small field inflation model with a Higgs-like potential does
not. We also find that an inflationary solution in a modified gravity theory
with limited curvature invariants is free of such a singularity and is smoothly
extendible.
