  In this paper, we study functions of bounded variation on a complete and
connected metric space with finite one-dimensional Hausdorff measure. The
definition of BV functions on a compact interval based on pointwise variation
is extended to this general setting. We show this definition of BV functions is
equivalent to the BV functions introduced by Miranda. Furthermore, we study the
necessity of conditions on the underlying space in Federer's characterization
of sets of finite perimeter on metric measure spaces. In particular, our
examples show that the doubling and Poincar\'e inequality conditions are
essential in showing that a set has finite perimeter if the codimension one
Hausdorff measure of the measure-theoretic boundary is finite.
