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Given a positive continuous function (except on closed set, where is zero ) on a smooth manifold how to find a smooth function under the same conditions being less (or equal) than this one continuous?

The hint was using a exhaustion of the complement of this closed set but I couldn't use it.

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The complement of the closed set is an open submanifold of your given manifold. Then apply Georges's solution here. This is only a start, probably you need some more cleverness (that's probably where the exhaustion comes in) to get the function smooth on the whole manifold. – t.b. Jul 13 '12 at 23:19

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