  The recent theory of graph limits gives a powerful framework for
understanding the properties of suitable (convergent) sequences $(G_n)$ of
graphs in terms of a limiting object which may be represented by a symmetric
function $W$ on $[0,1]$, i.e., a kernel or graphon. In this context it is
natural to wish to relate specific properties of the sequence to specific
properties of the kernel. Here we show that the kernel is monotone (i.e.,
increasing in both variables) if and only if the sequence satisfies a
`quasi-monotonicity' property defined by a certain functional tending to zero.
As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of
the form $W_1-W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its
own right; no such inequality holds for general kernels.
