  Given a hereditary property $\mathcal H$ of graphs and some $p\in[0,1]$, the
edit distance function $\operatorname{ed}_{\mathcal H}(p)$ is (asymptotically)
the maximum proportion of "edits" (edge-additions plus edge-deletions)
necessary to transform any graph of density $p$ into a member of $\mathcal H$.
For any fixed $p\in[0,1]$, $\operatorname{ed}_{\mathcal H}(p)$ can be computed
from an object known as a colored regularity graph (CRG). This paper is
concerned with those points $p\in[0,1]$ for which infinitely many CRGs are
required to compute $\operatorname{ed}_{\mathcal H}$ on any open interval
containing $p$; such a $p$ is called an accumulation point. We show that, as
expected, $p=0$ and $p=1$ are indeed accumulation points for some hereditary
properties; we additionally determine the slope of $\operatorname{ed}_{\mathcal
H}$ at these two extreme points. Unexpectedly, we construct a hereditary
property with an accumulation point at $p=1/4$. Finally, we derive a
significant structural property about those CRGs which occur at accumulation
points.
