I'm looking for an example of distribution of $(X,Y)$ with density $f_{XY}$ such that $\mathbb{E}[X|Y=y]$ is not continuous, but the marginal density $f_Y = \int f_{XY}$ is continuous.
In Transference of properties from marginals to joint density functions, there is a very nice example, when $f_{XY}(x,y)=1\{|x|+|y|\leq 1\}/2$ is discontinuous on $[-1,1]^2$, while $f_Y(y)=(1-|y|)$ is continuous. But in that example the conditional mean $$\mathbb{E}[X|Y=y]=\frac{1}{2(1-|x|)}\int_{-(1-|x|)}^{1-|x|}ydy=0 $$ is continuous.
Is there a good counterexample (or a way to show that it does not exist)?
Let $Y\sim U(-1,1)$ and $X\mid Y=y\sim N(1\{y>0\}, 1)$. Then
$$ \mathsf{E}[X\mid Y=y]=1\{y>0\}, $$ $$ f_X(x)=\frac{1}{2}\phi(x)+\frac{1}{2}\phi(x-1), \quad\text{and}\quad f_Y(y)=1/2. $$