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Mult. analysis

  1. Apr 5, 2009 #1
    Hello,I need some advice on a problem.

    Let [tex]f,g:R\rightarrow R[/tex] (where [tex]R[/tex] denotes the real numbers) be two continuous functions, assume that [tex]f(x) < g(x) \forall x \neq 0[/tex] ,

    and f(0) = g(0).Define [tex]A = \left\{(x,y)\neq (0,0): y< f(x),x \in R\right\}
    [/tex]

    [tex]B = \left\{(x,y)\neq (0,0): y> g(x),x \in R\right\}
    [/tex]


    Let [tex]r,s:R^{2}\rightarrow R [/tex] be [tex]C^{1}[/tex],and show that there is a [tex]C^{1}[/tex] function h defined on

    [tex]R^{2}- \left\{(0,0)\right\}[/tex] such that h(x,y) = r(x,y) on A and h(x,y) = s(x,y) on B.

    It seems that this follows from partitions of unity, but Im not sure where to
    start, any suggestions at all are helpful.
    Thanks.
     
    Last edited: Apr 5, 2009
  2. jcsd
  3. Apr 6, 2009 #2
     
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