  By extrapolating the explicit formula of the zero-bias distribution occurring
in the context of Stein's method, we construct characterization identities for
a large class of absolutely continuous univariate distributions. Instead of
trying to derive characterizing distributional transformations that inherit
certain structures for the use in further theoretic endeavours, we focus on
explicit representations given through a formula for the density- or
distribution function. The results we establish with this ambition feature
immediate applications in the area of goodness-of-fit testing. We draw up a
blueprint for the construction of tests of fit that include procedures for many
distributions for which little (if any) practicable tests are known. To
illustrate this last point, we construct a test for the Burr Type XII
distribution for which, to our knowledge, not a single test is known aside from
the classical universal procedures.
