property of the measurable function from conditional expectation

Question

Let $(X,Y)$ be a pair of integrable random variables. We know for the conditional expectation $\mathbb E(X|Y)$, there is a measurable function $f$ such that $\mathbb E(X|Y)=f(Y)$. I would like to investigate property of the function $f$: namely, are there any generic condition on the distribution of $(X,Y)$ such that

1. f is monotonically increasing?
2. f is continuous, or even smooth (assuming the support of $Y$ is an interval for example)?
3. f is linear? etc...

Necessary and sufficient conditions are certainly most desirable, but any idea and suggestion are greatly welcome!

EDIT： One sufficient condition for $f$ to be increasing is that $supp(\pi)$ is a monotone set, where $\pi$ is the distribution of $(X,Y)$. That is, for any $x,y$ belonging to the support of $\pi$, $(x_1-y_1)(x_2-y_2)\geq 0$. Apparently this is not a necessary condition which is not so satisfactory.