Consider the following family of measures on $(\mathbb R,\mathcal B(\mathbb R))$: $ K_x(A) = \begin{cases} \int\limits_A \frac{1}{|x|\sqrt{2\pi}}\mathrm e^{-y^2/2x^2}\,dy&,\text{ if }x\neq 0, \\ I_{A}(0)&,\text{ if }x = 0. \end{cases} $ where $I_A(t)$ is an indicator (characteristic) function of the set $A$. I wonder if it is possible to find a measure $\mu$ such that $\mu>>K_x$ for any $x$ (i.e. if $\mu(A) = 0$ then $K_x(A) = 0$) and $ \xi(x,y):=\frac{\mathrm dK_x}{\mathrm d\mu} $ is a continuous function of $(x,y)$.
Of course, we cannot take $\mu$ to be Lebesgue measure since $K_0(\{0\}) = 1$ and the only problem is in this point. Still I am not sure that there are no such measures at all, but I didn't fund an example or prove that there are no an example.