Continuous selection given both upper and lower hemicontinuity

Suppose that $X,Y$ are compact metric spaces, and $g:X\rightarrow Y$ is a multivalued mapping that is both upper and lower hemicontinuous. Is there a single valued continuous selection of $g$? If it helps, $X$ and $Y$ are compact subsets of finite dimensional normed spaces, but $g(x)$ is not convex, in general.

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Since Michael's theorem does not apply, is there some other well known result in set-valued analysis that covers this situation? A reference would be greatly appreciated. – Brian Lins Aug 7 '12 at 19:23