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Extension of ๐ถ^{โˆž} functions defined in a half space

Extension of ๐ถ^{โˆž} functions defined in a half space

The following question arises in connection with manifolds with boundary: given a function f defined and Co in a half space, and all of whose derivatives have continuous limits at the boundary, can f be extended to a Coo function in the whole space? More specifically, let xERn, tER, S+ โ€ฆ