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I am having troubles with the concept of hemicontinuity. In particular, since I am studying Econ, we did not treated it properly but we are supposed to know how to use it. Then I think this question will be very basic for most of you but it can be of help in making me understanding.

I asked for the proof with sequence characterization because it is the one we are supposed to use.

From my understanding, in layman words, upper hemicontinuity means that any sequence in the correspondence converges to a point in the correspondence; while, lower hemicontinuity means that every point in the correspondence can be reached by a sequence in the correspondence.

Now, let's assume we have a correspondence defined as

\begin{gather} \begin{cases} [.3,.7]\,\,\ if\,\,\ x \leq \frac{1}{2} \\ \{\frac{1}{2}\}\,\,\,\,\,\,\ if\,\,\ x >\frac{1}{2} \end{cases} \end{gather}

whose graph is reported below. I am said that this correspondence is uhc but not lhc at $\frac{1}{2}$. How should I proceed?

enter image description here

Source: Notes from UArizona

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