I have just been introduced to the concept of "hemi-continuity" (the "h" is not a typo) of correspondences.
If I understand the concept correctly then the following conjecture should be true:
A correspondence that maps $x\to \{f(x)\}$ (i.e. a single valued correspondence), is lower (or upper) Hemi-continuous iff $f$ is lower (or upper) Semi-continuous
Is my conjecture correct? This is just my first attempt at connecting hemi-continuity to what I already understand.
No. An upper hemi-continuous correspondence has a closed graph, but a semi-continuous doesn't. They are related in the sense that $\leq$ over real numbers is similar to $\subseteq$ over subsets.