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Suppose $n,m\geq 1$ and let $A\subset \mathbb{R}^m$ compact. Let $f:\mathbb{R}^n\times A\to \mathbb{R}^n$ and $l:\mathbb{R}^n\times A\to \mathbb{R}$ such that

  • $f$ is continuous and there exist $L_f,M_f>0$ such that $$\left\lVert f(x_1,a)-f(x_2,a)\right\rVert \leq L_f\left\lVert x_1-x_2\right\rVert,\qquad\left\lVert f(x_1,a)\right\rVert \leq M_f$$ for all $x_1,x_2 \in \mathbb{R}^n$ and $a\in A$.

  • $l$ is continuous, and there exist a modulus of continuity $w_l$ and $M_l>0$ such that $$|l(x_1,a)-l(x_2,a)| \leq w_l(\left\lVert x_1-x_2\right\rVert)\qquad |l(x_1,a)|\leq M_l$$ for all $x_1,x_2 \in \mathbb{R}^n$ and $a\in A$.

Fix $\lambda>0$. I have to prove that the function $F:\mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}$ such that $$F(x,z,p)=\lambda z+\sup_{a \in A}\{-p\cdot f(x,a)-l(x,a)\}$$ for all $(x,z,p)\in \mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^n$ is continuous.

Certainly $F$ is well posed since for every $(x,p)$ we have $$ \left|\sup_{a \in A}\{-p\cdot f(x,a)-l(x,a)\}\right|\leq (\left\lVert p\right\rVert M_f+M_l).$$

I'm a little bit confused about how to proceed in order to prove continuity.

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For $x,p\in\mathbb R^n$ and $a\in A$, denote $H(x,p,a)=-p\cdot f(x,a)-l(x,a)$ and $G(x,p)=\sup\{H(x,p,a)\!:a\in A\}$. Let $x_1,p_1\in\mathbb R^n$ and $\varepsilon>0$ be fixed. Then there exists $\delta>0$ such that $$\big(\!\left\|p_1\right\|L_f+M_f\big)\delta+\omega_l(\delta)\le\varepsilon.$$ If $\left\|x_1-x_2\right\|\le\delta$ and $\left\|p_1-p_2\right\|\le\delta$ then for all $a\in A$ we have $$\big|H(x_1,p_1,a)-H(x_2,p_2,a)\big|\le\big|p_1\cdot f(x_1,a)-p_1\cdot f(x_2,a)\big|+\big|p_1\cdot f(x_2,a)-p_2\cdot f(x_2,a)\big|+\big|l(x_1,a)-l(x_2,a)\big|\le\left\|p_1\right\|\big\|f(x_1,a)-f(x_2,a)\big\|+\left\|p_1-p_2\right\|\big\|f(x_2,a)\big\|+\omega_l(\left\|x_1-x_2\right\|)\le\big(\!\left\|p_1\right\|L_f+M_f\big)\delta+\omega_l(\delta)\le\varepsilon,$$ hence $H(x_1,p_1,a)\le G(x_2,p_2)+\varepsilon$ and $H(x_2,p_2,a)\le G(x_1,p_1)+\varepsilon$. It folows that $G(x_1,p_1)\le G(x_2,p_2)+\varepsilon$ and $G(x_2,p_2)\le G(x_1,p_1)+\varepsilon$, hence $\big|G(x_1,p_1)-G(x_2,p_2)\big|\le\varepsilon$.

This implies the continuity of $G$. The continuity of $F$ follows easily.

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