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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.

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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:

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..

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  • $\begingroup$ Thank you for your references. All the references you provided are the definitions for the hemicontinuity of set-valued mappings. Where can I find the source for single valued functions? $\endgroup$
    – Blind
    Feb 1 '15 at 0:25
  • $\begingroup$ I'm looking, but it seems hard to find. I think that this might be closer to what you need. $\endgroup$
    – Ivo Terek
    Feb 1 '15 at 0:32
  • $\begingroup$ Thank you for your great help. In your reference, they define the hemicontinuity for mapping $T:X\rightarrow X^*$. Here, we want the definition for mapping $f:X\rightarrow \mathbb{R}$. $\endgroup$
    – Blind
    Feb 1 '15 at 0:37
  • $\begingroup$ Damn, I'm trying haha. Here the definition in page $2$ is for maps into $\Bbb R$. I'll keep looking for more. $\endgroup$
    – Ivo Terek
    Feb 1 '15 at 0:41
  • $\begingroup$ I knew this paper. The definition in this paper also works for mapping $A: X\rightarrow X^{\prime}$. $\endgroup$
    – Blind
    Feb 1 '15 at 0:44

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