Law of Iterated Expectation, Knowing the odds I am reading knowing the odds and have some questions about Proposition $3.41$ which is law of iterated expectation.

Proposition $3.41$ 
Let $X$ and $Y$ be random variables. Suppose that they have a continuous joint density $f_{XY}(x,y)$ and that $X$ has a strictly positive continuous density $f_{X}(x)$. If $Y$ is integrable, then $E(Y|X)$ is also integrable and $$E(E(Y|X))=E(Y)$$

Here's the proof

Let us suppose $Y$ is positive, so that it's integrable iff $E(Y)$ is finite and write $\phi(X)=E(Y|X)$
$$E(E(Y|X))=E(\phi(X))= \int_{-\infty}^{\infty}\phi(x)f_{X}(x)dx
=\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}yf_{Y|X}(y|x)dy\right)f_{X}(x)dx$$
$$=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}y \frac{f_{XY}(x,y)}{f_{X}(x)}dyf_{X}(x)dx$$
  Interchange the order and we finally get $E(Y)$

My questions are 
$(1)$ why do we need to assume $X$ has strictly positive continuous density? It seems as long as it's not zero, we can get the result. 
$(2)$The same question for $Y$, why do we discuss whether $Y$ is positive or not? 
$(3)$What role does continuity play in the whole proof? Both continuity of joint density and marginal density
Thanks in advance
 A: You wrote:

Let us suppose $Y$ is positive, so that it's integrable iff $E(Y)$ is finite

The point is that integrability of $Y$ means that $\operatorname E(|Y|) < +\infty,$ and that's the same as saying $\operatorname E( Y \cdot \mathbf 1_{Y\,\ge\, 0})<+\infty$ and $\operatorname E(-Y\cdot\mathbf 1_{Y\,<\,0}) < +\infty.$ Assuming $Y$ is positive means you don't need to think about the second item here, but only the first.
Positivity of the density of $X$ just assures that you can divide by it. In this context, saying it's not zero is the same as saying it's positive, because density functions are never negative. (A complication here is that density functions are not defined pointwise: changing the value of a function at a point does not alter its integral, and the only thing that matters about density functions is what their integrals are over measurable sets. Being positive therefore means only that its integral over every open interval is positive.)
The density functions themselves were not assumed to be continuous. Rather it is assumed that the probability distributions are continuous distributions, which in this context means just that density functions exist, so that the probability that a random variable lies within any particular region is the integral of the density over that region.
