  This paper reviews a class of univariate piecewise polynomial functions known
as discrete splines, which share properties analogous to the better-known class
of spline functions, but where continuity in derivatives is replaced by (a
suitable notion of) continuity in divided differences. As it happens, discrete
splines bear connections to a wide array of developments in applied mathematics
and statistics, from divided differences and Newton interpolation (dating back
to over 300 years ago) to trend filtering (from the last 15 years). We survey
these connections, and contribute some new perspectives and new results along
the way.
