What is condition of set C where Point-wise supremum

Question

In convex optimization,

$$ g=\underset{y\in C}{\mathop{\sup }}\,f(x,y) $$

where $f$ is convex in $x$ for each $y$, $y$ is belong to $C$.

we know that $g(x)$ is convex in $x$

I have two questions which are associated each other

1. what should $C$(set of $y$) satisfy condition for $g$ is convex of $x$

2. In case of $g=\underset{y\in \{y\left| y\le h(x)\} \right.}{\mathop{\sup }}\,\,\,f(x,y)$ , this $g$ is also convex????

Answer 1

The set $C$ may not depend on $x$. Otherwise, even for a discrete set $C$, the function $h$ is convex since the supremum of convex functions is convex and that for fixed $y$, the function $h_y(x) = f(x,y)$ is convex.