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Lower Hemicontinuity

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Please determine whether or not the 2 correspndences is lower hemicontinuous and please justify why (using definition/proof of lower hemicontinuity):

1)F:R^2->R^2, F(u)={x: x o u =0}

2)F:R^n{0}->R^n, F(x)=B(x;||x||), the closed ball centred at x with radius ||x||.

Note: o is the dot product
is the complement
||x|| is the distance

My definiton of lower hemicontinuity is:

F:A->B is a correspondence, A is contained in R^n, B is contained in R^k, p belongs to A. I say that F is lowerhemicontinuous at p if: for all sequences {x^m} in A with x^m->p and for all q belonging to F(p), there exists {y^m} in B with y^m belonging to F(x^m) for all m and y^m->q.

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Solution Summary

Lower Hemicontinuity is investigated. The solution is detailed and well presented. The expert examines dot products for complements.

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1. ,
Proof: We consider a sequence and . Then we consider an arbitrary point , then we have . This implies and for some real number . Now we ...

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