Let $Y$ and $W$ be two jointly distributed random variables; $Y$ takes values on $(y_1,y_2)$ and $W$ takes values on $(w_1,w_2)$. The conditional probability density of $W$ given $Y$ is given by $f_{W|Y}$, assumed to be continuous and twice differentiable.

Let $X$ be a continuous increasing function of random variable $W$. The expected value of $X$ conditional on $Y=y$ is given by:

$$\tilde{X}(y)\equiv\mathrm{E}_{W|Y}[X(W)|Y=y]=\int_{w_1}^{w_2} X(w)f_{W|Y}(w|y)\mathrm{d}w $$

Which condition must be imposed on $f_{W|Y}$ in order to have $\tilde{X}(y)$ finite and $\tilde{X}^{\prime}(y)>0$ for every $y$?

EDIT: I conjecture that if $f_{W|Y}$ is such that $g(y)\equiv\mathrm{E}_{W|Y}[W|Y=y]$ is increasing, then the result follows. But I could not show that formally in terms of fundamental conditions about $f_{W|Y}$.