For two continuous functions $f$ and $g$ defined on a normed space $E$ taking values in $[-\infty,\infty)$, let $f\square g: x\mapsto \inf\{f(y)+g(x-y): y \in E\}$. Here I do not assume any properties of $f$ and $g$ other than continuity.
Is the function $f\square g$ necessarily continuous?
I am particularly interested in the case where $E$ is the set of bounded measurable functions on some measurable space (i.e. bounded random variables).
I know two facts:
1) if one of $f$ and $g$ is upper-semicontinuous then so is $f\square g$.
2) if one of $f$ and $g$ is uniformly continuous then so is $f\square g$.
However for continuity it seems very different and I could not find a counter-example.