  We use blowup to study the regularization of codimension one two-fold
singularities in planar piecewise smooth (PWS) dynamical systems. We focus on
singular canards, pseudo-equlibria and limit cycles that can occur in the PWS
system. Using the regularization of Sotomayor and Teixeira \cite{Sotomayor96},
we show rigorously how singular canards can persist and how the bifurcation of
pseudo-equilibria is related to bifurcations of equilibria in the regularized
system. We also show that PWS limit cycles are connected to Hopf bifurcations
of the regularization. In addition, we show how regularization can create
another type of limit cycle that does not appear to be present in the original
PWS system. For both types of limit cycle, we show that the criticality of the
Hopf bifurcation that gives rise to periodic orbits is strongly dependent on
the precise form of the regularization. Finally, we analyse the limit cycles as
locally unique families of periodic orbits of the regularization and connect
them, when possible, to limit cycles of the PWS system. We illustrate our
analysis with numerical simulations and show how the regularized system can
undergo a canard explosion phenomenon.
