  We examine four different notions of variation for real-valued functions
defined on the compact ring of integers of a non-Archimedean local field, with
an emphasis on regularity properties of functions with finite variation, and on
establishing non-Archimedean Koksma inequalities. The first version of
variation is due to Taibleson, the second due to Beer, and the remaining two
are new. Taibleson variation is the simplest of these, but it is a coarse
measure of irregularity and it does not admit a Koksma inequality. Beer
variation can be used to prove a Koksma inequality, but it is order-dependent
and not translation invariant. We define a new version of variation which may
be interpreted as the graph-theoretic variation when a function is naturally
extended to a certain subtree of the Berkovich affine line. This variation is
order-free and translation invariant, and it admits a Koksma inequality which,
for a certain natural family of examples, is always sharper than Beer's.
Finally, we define a Fourier-analytic variation and a corresponding Koksma
inequality which is sometimes sharper than the Berkovich-analytic inequality.
