I'm now approaching for the first time to the definition of upper envelope of a family of functions, and I just wonder to know some basic properties.
Suppose $\Omega $ is an open subset of $\mathbb{R}^n$. Let $A$ a family of index. For each $a\in A$ we have two functions $f_a:\Omega\to \mathbb{R}^n$, $l_a:\Omega\to\mathbb{R}$ and a scalar number $c_a\in \mathbb{R}$.
For each fixed $a$ we consider the real valued function: $$g_a:\Omega\times\mathbb{R}\times\mathbb{R}^n\to \mathbb{R}$$ such that $$g_a(x,z,p):=c_a z+f_a(x)\cdot p+l_a(x)$$ for all $(x,z,p)\in \Omega\times\mathbb{R}\times\mathbb{R}^n$, where $(\cdot)$ is the usual scalar product on $\mathbb{R}^n$.
We can now consider the upper envelope of the family $(g_a)_{a\in A}$, namely $$F(x,z,p):=\sup_{a\in A}g_a(x,z,p)=\sup_{a\in A}\{c_a z+f_a(x)\cdot p+l_a(x)\}.$$
My question is the following:
What are sufficient conditions to have:
1) $|F(x,z,p)|\neq \infty$ for every $(x,z,p)$
2) $F$ is continuous on $\Omega\times\mathbb{R}\times\mathbb{R}^n$??
Any suggestions will be really appreciated.
I would suggest the following conditions:
1) $c_a$ is bounded, and $f_a(x)$ and $l_a(x)$ as well for each $x$.
2) $f_a$ and $l_a$ are locally uniformly bounded, ie for all $x \in \Omega$, there exists some neighborhood $U$ of $x$ and some $M_U > 0$ such that for every $y \in U$, $|f_a(y)| +|l_a(y)| \leq M_U$.
3) The $f_a$ and $l_a$ are equicontinuous, ie: for all $x \in \Omega$ and $\epsilon >0$, there exists some neighborhood $U \subset \Omega$ of $x$ such that for all $a \in A$, $y \in U$, $|f_a(x)-f_a(y)|+|l_a(x)-l_a(y)| \leq \epsilon$.
1) ensures that $F$ is well-defined.
2) ensures that each $g_a$ is locally (wrt $x$) Lipschitz-continuous wrt $(p,z)$ and that the constants and neighborhoods can be chosen independently from the index, thus the property goes on to $F$.
3) Gives the same kind of continuity wrt $x$.