# Lipschitz continuity of multivariable function in expected value

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

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 $$K=L_fL_h$$.