  Let $X$ be a finite dimensional compact metrizable space. We study a
technique which employs semiprojectivity as a tool to produce approximations of
$C(X)$-algebras by $C(X)$-subalgebras with controlled complexity. The following
applications are given. All unital separable continuous fields of C*-algebras
over $X$ with fibers isomorphic to a fixed Cuntz algebra $\mathcal{O}_n$,
$n\in\{2,3,...,\infty\}$ are locally trivial. They are trivial if $n=2$ or
$n=\infty$. For $n\geq 3$ finite, such a field is trivial if and only if
$(n-1)[1_A]=0$ in $K_0(A)$, where $A$ is the C*-algebra of continuous sections
of the field.
  We give a complete list of the Kirchberg algebras $D$ satisfying the UCT and
having finitely generated K-theory groups for which every unital separable
continuous field over $X$ with fibers isomorphic to $D$ is automatically
(locally) trivial.
  In a more general context, we show that a separable unital continuous field
over $X$ with fibers isomorphic to a $KK$-semiprojective is trivial if and only
if it satisfies a K-theoretical Fell type condition.
