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Math Help - Darboux Theroem

  1. #1
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    Wink Darboux Theroem

    Let K be a D-domain and let f:K-->R be differentiable on the interval [a,b] as they are a subset of K. (a<b)

    Let x be an element of (a,b). Show that lim (y-->x+) f'(y) and lim (y-->x-) f'(y) both exist then f' must be continuous at x.

    Any help in solving this using Darboux would be greatly appreciated.
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  2. #2
    ynj
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    Quote Originally Posted by derek walcott View Post
    Let K be a D-domain and let f:K-->R be differentiable on the interval [a,b] as they are a subset of K. (a<b)

    Let x be an element of (a,b). Show that lim (y-->x+) f'(y) and lim (y-->x-) f'(y) both exist then f' must be continuous at x.

    Any help in solving this using Darboux would be greatly appreciated.
    According to Darboux Theorem, f'(x)have no discontinuity point of first kind. So \lim_{y\rightarrow x^{-}}f'(y)=\lim_{y\rightarrow x^{+}}f'(y)=f'(x), so we are done.
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