  The border-collision normal form describes the local dynamics in continuous
systems with switches when a fixed point intersects a switching surface. For
one-dimensional cases where the bifurcation creates or destroys only fixed
points and period-two orbits, we show that the standard local equivalence of
normal forms, topological conjugacy, can be replaced by differentiable
conjugacy provided an extra term is added to the normal form. In these cases
topological conjugacy is so weak that a range of values can be used for the
coefficients in the normal form. The extension to differentiable conjugacy
explains why the usual choice of slopes in the standard normal form is
privileged. This highlights the importance of differentiable conjugacies and
the need for extended normal forms.
