Suppose $h:\mathcal{X \times Y \times W}\rightarrow\mathcal{X}$, with $\mathcal{X,Y,W} \subseteq \mathbb R^d$, $d \in \mathbb N$ is Lipschitz in $x,y$, i.e., 
$$ \| h(x,y,w) - h(x',y',w) \|\_2 \le L\_h (\|x - x' \|\_2 + \| y -y'\|\_2) \quad \forall \, x,x' \in \mathcal X, \ \forall \, y,y'\in \mathcal Y. $$ 
We also have $f: \mathcal X \rightarrow \mathbb R$ is Lipschitz in $x$, i.e.,
$$ | f(x) - f(x') | \le L\_f \|x - x' \|\_2 \quad \forall \, x,x' \in \mathcal X.$$ 
Can we say that 
$$ \max\_y \left|\int\_{\mathcal W} p(w) f(h(x,y,w)) dw - \int\_{\mathcal W} p(w) f(h(x',y,w)) dw \right| \le K \|x-x' \|\_2 \quad \forall \, x,x' \in \mathcal X,$$
(where $p(w)$ is a probability density function), i.e., $\int\_{\mathcal W} p(w) f(h(x,y,w)) dw$ is Lipschitz in $x$? If yes what is the Lipschitz constant $K$?


We can assume for example that $h(x,y,w)= g(x,y) + w$ where $g:\mathcal{X\times Y}\rightarrow\mathcal{X}$ is a deterministic function of $x,y$ and $w \sim \mathcal N(\boldsymbol{\mu}, \Sigma)$. Thanks



Lipschitz continuity of multivariable function in expected value
Assume that 
$$|f(x)-f(x')|\le L\_f\|x-x'\|\_2\quad\forall \ x,x' \in \mathcal X,$$
rather than 
$$|f(x)-h(x')|\le L\_f\|x-x'\|\_2\quad\forall \ x,x' \in \mathcal X.$$
Then 
$$|f(h(x,y,w))-f(h(x',y,w))|\le L\_f\|h(x,y,w)-h(x',y,w)\|\_2
\le L\_fL\_h\|x-x'\|\_2,$$
whence 
$$\left|\int\_{\mathcal W} p(w) f(h(x,y,w))\, dw - \int\_{\mathcal W} p(w) f(h(x',y,w)) \,dw \right| \\ 
\le\int\_{\mathcal W} p(w) |f(h(x,y,w))-f(h(x',y,w))| dw \\ 
\le\int\_{\mathcal W} p(w) L\_fL\_h\|x-x'\|\_2 dw = L\_fL\_h\|x-x'\|\_2.$$
So, your desired inequality holds with $K=L\_fL\_h$.


