I want an idea to prove whether $$ f(x_1)=g(x_1)-a E[\max\{x_1,x_2\}]-bE[\max\{x_1-c,0\}] $$ is uhc (continuous) in $x_1$ and $x_2$ given that $g(x_1)$ is strictly concave and $a$, $b$, and $c$ are constants. Please offer some suggestions.
upper hemicontinuous (uhc)
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calculus
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2Yes. These tags make a lot of sense. – 2012-08-29
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0Is there a difference between "hemicontinuity" and what is usually called "semicontinuity"? "Hemi-" is to "semi-" as "hexa-" to "sexa-" and "hyper-" to "super-" and "hypo-" to "sub-", etc.: the first is Greek and the second Latin. Latin seems to fit better with a word like "continuous". – 2012-08-29
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1@MichaelHardy [Upper hemicontinuity](http://en.wikipedia.org/wiki/Hemicontinuity) is a notion of continuity for set-valued mappings or correspondences. People sometimes use *semicontinuous* for the same concepz, semicontinuous is ambigous, since it might refer to a weakened continuity notion for functions. – 2012-08-29
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This is actually a function, so continuity and upper hemicontinuity coincide. This function is easily seen to be continuous, once one uses the nontrivial fact that a concave function defined on an open interval is also continuous on that interval and the fact that the max-function is continuous.