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Simple Proof of Weierstass Approximation Theorem?

  1. Mar 29, 2010 #1
    1. The problem statement, all variables and given/known data
    Let D={x in the set of real numbers: -3<x<3, x does not equal 0,1,2} and define g(x)=(cosx-1)/x + (x3-2x2-x+2)/(x2-3x+2) on D. Find G:R→R such that G is continuous everywhere and G(x)=g(x) when x is in set D.

    2. Relevant equations

    3. The attempt at a solution

    From a past homework problem I know how to prove that, for any continuous f:R→R, there exists a sequence (pn) of polynomials such that pn converges uniformly to f on any given bounded subset of R.
    So after I show that g(x) is continuous and that a sequence of polynomials that converges uniformly to g exists, how do I actually find the function G(x)?
  2. jcsd
  3. Mar 29, 2010 #2
    Use the definition of continuity: g(0) = lim(x->0) g(x) = 1 etc. (The limits of the form 0/0 are to be tackled with L'Hospital's rule).
    You might want to have a look at Riemann's theorem on removable singularities to see how this is done in general.
    P.S. - Where is the Weirstrass Approximation Thm. called for?
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