References for hemicontinuity?

Question

Let $X$ be a real vector space, $K\subset X$ be a nonempty and convex set. The mapping $f:X\rightarrow\mathbb{R}$ is said to be hemicontinuous if for every $u,v\in K$, the mapping $g(t):[0,1]\rightarrow\mathbb{R}$ given by $g(t)=f(tu+(1-t)v)$ is continuous.

I would like to find references and properties for this function.

Thank you for all kind help and comments.

Answer 1

In these notes there are a few comments in page $5$:

What should it mean for a set to “jump down” at the limit $x_0$? It should mean the set suddenly gets smaller – it “implodes in the limit” – that is, there is a sequence $x_n → x_0$ and points $y_n \in \Psi(x_n)$ that are far from every point of $\Psi(x_0)$ as $n → ∞$. (See Figure 4.) Similarly, what should it mean for a set to “jump up” at the limit? This should mean that that the set suddenly gets bigger – it “explodes in the limit” – that is, there is a point $y \in \Psi(x_0)$ and a sequence $x_n → x_0$ such that $y$ is far from every point of $\Psi(x_n)$ as $n → ∞$.

which try to justify the definition. The figures are in page $11$.

There's always good an' old wikipedia, too:

Roughly speaking, a function is upper hemicontinuous when (1) a convergent sequence of points in the domain maps to a sequence of sets in the range which (2) contain another convergent sequence, then the image of limiting point in the domain must contain the limit of the sequence in the range. Lower hemicontinuity essentially reverses this, saying if a sequence in the domain converges, given a point in the range of the limit, then you can find a sub-sequence whose image contains a convergent sequence to the given point.

More important than this comment, are the references given in the end of the page, which might be worth looking for:

• Jean-Pierre Aubin, Helene Frankowska: Set-Valued Analysis, Birkhäuser, Basel, 1990
• Klaus Deimling: Multivalued Differential Equations, Walter de Gruyter, 1992
• Charalambos D. Aliprantis, Kim C. Border: Infinite dimensional analysis. Hitchhiker's guide, Springer, 1994

You can look around MSE itself, it seems that are a few questions about the subject, e.g. this one.

There's also some notes here (about page $21$), which seems to be based in the first link I gave you, and here.

I think you may be good for a start now..