  Structural properties are given for $D(K)$, the Banach algebra of (complex)
differences of bounded semi-continuous functons on a metric space $K$. For
example, it is proved that if all finite derived sets of $K$ are non-empty,
then a complex function $\varphi$ operates on $D(K)$ (i.e., $\varphi\circ f\in
D(K)$ for all $f\in D(K)$) if and only if $\varphi$ is locally Lipschitz.
Another example: if $W\subset K$ and $g\in D(W)$ is real-valued, then it is
proved that $g$ extends to a $\tilde g$ in $D(K)$ with $\|\tilde g\|_{D(K)} =
\|g\|_{D(W)}$. Considerable attention is devoted to $SD(K)$, the closure in
$D(K)$ of the set of simple functions in $D(K)$. Thus it is proved that every
member of $SD(K)$ is a (complex) difference of semi-continuous functions in
$SD(K)$, and that $|f|$ belongs to $SD(K)$ if $f$ does. An intrinsic
characterization of $SD(K)$ is given, in terms of transfinite oscillation sets.
Using the transfinite oscillations, alternate proofs are given of the results
of Chaatit, Mascioni and Rosenthal that functions of finite Baire-index belong
to $SD(K)$, and that $SD(K)\ne D(K)$ for interesting $K$. It is proved that the
``variable oscillation criterion'' characterizes functions belonging to
$B_{1/4}(K)$, thus answering an open problem raised in earlier work of Haydon,
Odell and Rosenthal. It is also proved that $f$ belongs to $B_{1/4}(K)$ (if
and) only if $f$ is a uniform limit of simple $D$-functions of uniformly
bounded $D$-norm iff $\osc_\omega f$ is bounded; the last equivalence has also
been obtained by V.~Farmaki, using other methods.
