Let $(\theta_1,\theta_2) \in [0,1]^2$ be a two-dimensional random variable with a absolutely continuous joint distribution $F$ with full support. Can we say $\mathbb{E}(\theta_2|\theta_1)$ is continuous in $\theta_1$? A proof or a counterexample would be equally welcome.
My approach: $\mathbb{E}(\theta_2|\theta_1)=\int\limits^1_0\theta_2dF(\theta_2|\theta_1)=1-\int\limits^1_0F(\theta_2|\theta_1)$, integrating by parts. I know $F$ is continuous. But after this I'm confused if I can conclude $F(\theta_2|\theta_1)$ is continuous is $\theta_1$ and $\theta_2$.
Consider the following joint density of two random variables $X$ and $Y$: $$ f(x,y)=\cases{1, & $0\le x\le 1,0\le y<1/2$, \\ 2(1-x), & $0\le x\le 1,1/2\le y\le 1$, \\ 0, & otherwise.} $$ In this case $$ \mathsf{E}[X\mid Y=y]=\frac{1}{2}\times 1_{[0,1/2)}(y)+\frac{1}{3}1_{[1/2,1]}(y). $$