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Surjective Proof

  1. Apr 27, 2014 #1
    1. The problem statement, all variables and given/known data
    Suppose f: (a,b)→R where (a,b)[itex]\subset[/itex]R is an open interval and f is a differentiable function. Assume that f'(x)≠0 for all x[itex]\in[/itex](a,b). Show that there is an open interval (c,d)[itex]\subset[/itex]R such that f[(a,b)]=(c,d), i.e. f is surjective on (c,d).


    2. Relevant equations
    f is surjective if for all y[itex]\in[/itex]R there exists an x[itex]\in[/itex]X such that f(x)=y.


    3. The attempt at a solution
    I think I'm supposed to use ε and δ for this proof but I'm not sure where to start. Any clues would be great! Thanks.
     
  2. jcsd
  3. Apr 27, 2014 #2

    Dick

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    I don't think you need ε and δ. Start by thinking about continuous functions (like f, since it's differentiable) and the Intermediate Value Theorem. The Mean Value theorem will come in handy too.
     
    Last edited: Apr 27, 2014
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