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Thread: 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, $\displaystyle f'(x)$have no discontinuity point of first kind. So $\displaystyle \lim_{y\rightarrow x^{-}}f'(y)=\lim_{y\rightarrow x^{+}}f'(y)=f'(x)$, so we are done.
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