  We consider Hilbert spaces of analytic functions in the disk with a
normalized reproducing kernel and such that the backward shift $f(z) \mapsto
\frac{f(z)-f(0)}{z}$ is a contraction on the space. We present a model for this
operator and use it to prove the surprising result that functions which extend
continuously to the closure of the disk are dense in the space. This has
several applications, for example we can answer a question regarding reverse
Carleson embeddings for these spaces. We also identify a large class of spaces
which are similar to the de Branges-Rovnyak spaces and prove some results which
are new even in the classical case.
