# Upper semi-continuity

### From Glossary

(or *upper semi-continuous*, abbr. *usc*). Suppose
Of a function,
Of a point-to-set map, let
be a neighborhood of the set S. For each e > 0, there exists K such that for all
is a subset of
Here is an example of what can go wrong. Consider the feasibility map with

Note g is continuous and its level set is
However, for any

which is not bounded. The map fails to be usc at 0 due to the lack of stability of its feasibility region when perturbing its right-hand side (from above).