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Math Help - Subadditive Proof

  1. #1
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    Subadditive Proof

    Suppose that a function f is a subadditive prove that if f(0)=0 and if f is continuous at x=0 then f is continuous on all of R ?

    how can i solve it ?
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  2. #2
    Junior Member
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    A function f(x) is continuous at a point a if \lim_{x\to a}f(x)=f(a). Another characterization of continuity at a (which is useful in your case) is that \limsup_{x\to a}f(x)-\liminf_{s\to a}f(x)=0. ( \limsup of course is defined as \limsup_{x\to a}f(x)=\lim_{\epsilon\to0}\ \sup\{f(x):\lvert x-a\rvert<\epsilon,x\neq a\}

    I would use the subadditivity of f along with the above characterization to show continuity at an arbitrary point.
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