Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question about extentions of smooth functions

  1. Apr 9, 2008 #1
    My question is simple :

    Suppose that [tex]f[/tex] is in [tex]C^\infty(U, [0,1])[/tex] where [tex]U[/tex] is an open of [tex]R^n[/tex] .
    Is there [tex]g[/tex] in [tex]C^\infty(R^n,[0,1])[/tex] such that [tex]f=g[/tex] on [tex]U[/tex] ?

    I would say yes, but I don't know how to prove it.

    Thanks
     
  2. jcsd
  3. Apr 9, 2008 #2
    Did you try it for the special case n=1?

    Take for example U=(0,1) and f:U->[0,1] defined by f(x)=x.

    This satisfies your conditions, can you extend it to a C^\infty function on R...?

    I'd say no but the proof is left to you. :smile:

    If you were to replace [0,1] by [itex]\mathbb{R}[/itex], then I'd be with you that the assertion should hold true.
     
    Last edited: Apr 9, 2008
  4. Apr 10, 2008 #3
    OK, I guess I should add some hypothesis on f and change the conclusion in order to have something that could be true:

    Here is the new problem :

    Let [tex]\epsilon>0[/tex].
    Suppose that [tex]f[/tex] is in [tex]C^\infty(U, [0,1])[/tex] where [tex]U[/tex] is an open of [tex]R^n[/tex] and suppose that for any [tex] x_0 [/tex] in [tex]\partial U[/tex] (the boundary of U), and any n-multi-indice [tex]\alpha[/tex], the limit

    [tex]
    \lim_{x\in U, x\to x_0} \partial^\alpha f (x)
    [/tex]

    exists and is in [tex]R[/tex].


    Is there [tex]g[/tex] in [tex]C^\infty(R^n,[-\epsilon, 1+\epsilon])[/tex] such that [tex]f=g[/tex] on [tex]U[/tex] ?
     
  5. Apr 11, 2008 #4

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    i would try reflecting f to the other side of the ball U, then using that to extend f a little bit, then chopping f off by a smooth bump function.

    at least if U is a nice ball.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Question about extentions of smooth functions
Loading...