Despite of difference in their name, they seem to be the same definition. Some texts use the name "hemi-continuity", others use the other name; none of those notes clarify the difference between hemi-continuity.
A Caltech note says:
The term semicontinuity is used by many authors to mean hemicontinuity.
I wonder if this is correct in general.
$\endgroup$ 21 Answer
$\begingroup$A correspondence $\Gamma:X\rightrightarrows Y$ is UHC at $x\in X$ if for every open subset $O$ of $Y$ with $\Gamma(x)\subseteq O$, $\exists\;\delta>0$ such that $\Gamma(N_{\delta, X}(x))\subseteq O$.
If the correspondence is single-valued (i.e. $\Gamma(x)=\{f(x)\}$ for all $x\in X$), the definition reduces to the definition of Continuity:$f:X\rightarrow Y$ is continuous at $x\in X$ if for every open subset $O$ of $Y$ with $f(x)\in O$, $\exists\; \delta>0$ such that $f(N_{\delta, X}(x))\subseteq O$.
So you can say something more: every single valued correspondence is continuous (not just upper semi-continuous).
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