Unformatted text preview: { g n } on R such that { f n } converges uniformly, { g n } converges uniformly but { f n g n } does not converge uniformly on R . 6. Let φ,ψ : R 3 → R be continuously diﬀerentiable functions and deﬁne F : R 3 → R 3 by F ( x,y,z ) = ( φ ( x,y,z ) ,ψ ( x,y,z ) ,φ 2 ( x,y,z ) + ψ 2 ( x,y,z )) (a) Check whether or not the inverse function theorem applies to F at any point ( x ,y ,z ), i.e., check if F satisﬁes the hypothesis of the inverse function theorem at any point ( x ,y ,z ). (b) Suppose that F ( ~a ) = ~ b for some points ~a, ~ b ∈ R 3 . Explain geometrically why F does not have an inverse function from an open set V ⊂ R 3 containing ~ b to an open set U ⊂ R 3 containing ~a ....
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 Spring '11
 NA
 Continuous function, Inverse Function Theorem

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