1. Consider the correspondence f: $\mathbb{R}$ -> $\mathbb{R}$ defined by:

f(a) = {x $\in$ $\mathbb{R}$:x2 + x $\leq$ a2} for all a $\in$ $\mathbb{R}$.

Find the points where the correspondence is uhc and the points where it is lhc. Justify your conclusions exhaustively.

  1. For a correspondence f: $\mathbb{R}$ -> $\mathbb{R}$, define its graph as:

$\Gamma$f = {(x, y) $\in$ $\mathbb{R}$2: y $\in$ f(x)}.

Prove that if $\Gamma$f is an open subset of $\mathbb{R}$2, then f is lhc.

I know that for a function to be upper hemicontinuous, there must be a convergent sequence in the domain mapping to a sequence of sets in range which contain convergent sequences, and an image of the convergent sequence limit must contain the limit of the sequence in the range. Additionally, for a lower hemicontinuous, if the function has open graph Gr(f) it is lower hemicontinuous. Do I have the right criteria?

I would appreciate any tips as to how to get started on these two questions, as I am somewhat unsure as to how to start.

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    $\begingroup$ You have incorrect definition of lower hemicontinuity. Wikipedia gives the correct definition. en.wikipedia.org/wiki/Hemicontinuity The figure on the right shows that the two definitions are different (not open graph, lhc function). $\endgroup$ – Giskard Nov 16 '15 at 13:53
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    $\begingroup$ I also have a recommendation on how to solve your first problem: draw the graph of the function. $\endgroup$ – Giskard Nov 16 '15 at 13:54

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