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Math Help - Continuity on an interval

  1. #1
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    Continuity on an interval

    Show that if f is continuous on [0,infinity) and uniformly continuous on [a,infinity) for some positive constant a, then f is uniformly continuous on [0,infinity)
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by gixxer998 View Post
    Show that if f is continuous on [0,infinity) and uniformly continuous on [a,infinity) for some positive constant a, then f is uniformly continuous on [0,infinity)
    Since f is continuous on [0,a] it is uniformly continuous, so there exists \delta_1 such that |x-y|<\delta_1\implies|f(x)-f(y)|<\epsilon. Similarly, there exists \delta_2 for the interval [a,\infty).

    So taking \delta=\min\{\delta_1,\delta_2\} will work for [0,\infty).
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