  The article is devoted to investigation of classes of functions monotone as
functions on general $C^*$-algebras that are not necessarily the $C^*$-algebras
of all bounded linear operators on a Hilbert space as it is in classical case
of matrix and operator monotone functions. We show that for general
$C^*$-algebras the classes of monotone functions coincide with the standard
classes of matrix and operator monotone functions. For every class we give
exact characterization of $C^*$-algebras that have this class of monotone
functions, providing at the same time a monotonicity characterization of
subhomogeneous $C^*$-algebras. We use this characterization to generalize one
function based monotonicity conditions for commutativity of a $C^*$-algebra, to
one function based monotonicity conditions for subhomogeneity. As a
$C^*$-algebraic counterpart of standard matrix and operator monotone scaling,
we investigate, by means of projective $C^*$-algebras and relation lifting, the
existence of $C^*$-subalgebras of a given monotonicity class.
