The difference of two upper semi-continuous functions is in general neither upper- nor lower semi-continuous. But what can be said about the continuity set of such a functions, specifically its topological properties?
I know that the continuity set is a $G_\delta$ set.
On
Finally I found an useful result for my work. If $f$ is a real function over a polish space then $f$ is a Baire 1 function if and only if the restriction to a closed subset has a continuity point. That is the Baire Characterization Theorem. If a function is upper semi-continuous also it is Baire 1. The same for a difference of two upper semi-continuous real functions.
When you subtract, you throw upper semi-continuity properties out the window. This is because the negation could be very not-upper semi-continuous. For example, $$f(x) = \left\{\begin{array}{lr} 1 & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q}\end{array}\right.$$ is upper semi-continuous at each point in $\mathbb{Q}$. So is $$g(x) = 1$$ But $g - f$ is not upper semi-continuous at any point in $\mathbb{Q}$.
Restriction of an upper semi-continuous function behaves better. The set of points where the restricted function is upper semi-continuous contains all points where the original function was upper semi-continuous, as well as possibly some new points on the boundary of the restricted domain.