  We introduce a version of Stein's method of comparison of operators
specifically tailored to the problem of bounding the Wasserstein-1 distance
between continuous and discrete distributions on the real line. Our approach
rests on a new family of weighted discrete derivative operators, which we call
bespoke derivatives. We also propose new bounds on the derivatives of the
solutions of Stein equations for Integrated Pearson random variables; this is a
crucial step in Stein's method. We apply our result to several examples,
including the Central Limit Theorem, Polya-Eggenberger urn models, the
empirical distribution of the ground state of a many-interacting-worlds
harmonic oscillator, the stationary distribution for the number of genes in the
Moran model, and the stationary distribution of the Erlang-C system. Whenever
our bounds can be compared with bounds from the literature, our constants are
sharper.
