By Cheeger-Colding (see [CC00], theorem 4.6), in the regular set $\mathcal{R}$ of a general Ricci-limit space, the density function $\mathscr{V}_{\infty}$ of the renormalized limit measure always exists and is Hölder continuous. Our example tells us that, in general, one cannot expect the regularity of $\mathscr{V}_{\infty}$ to be differentiable in $\mathcal{R}$ (even though $\mathcal{R}$ is a smooth Riemannian manifold). We thank Shouhei Honda for pointing this out.