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Math Help - Show a function is continuous at every point in Reals

  1. #1
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    Show a function is continuous at every point in Reals

    Let K be greater than 0 and let f : Real numbers --> Real numbers satisfy the condition |f(x)-f(y)| less than or equal to K|x-y| for all x, y in Reals.
    Show that f is continuous at every point in c in Reals.
    Thank you.
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  2. #2
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    Quote Originally Posted by gixxer998 View Post
    Let K be greater than 0 and let f : Real numbers --> Real numbers satisfy the condition |f(x)-f(y)| less than or equal to K|x-y| for all x, y in Reals.
    Show that f is continuous at every point in c in Reals.
    \varepsilon  > 0\, \Rightarrow \,\delta  = \frac{\varepsilon }{2K}
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  3. #3
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    Quote Originally Posted by Plato View Post
    \varepsilon  > 0\, \Rightarrow \,\delta  = \frac{\varepsilon }{2K}
    Thank you. How did you come up with this?
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  4. #4
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    that's Lipschitz condition, and every Lipschitz function is uniformly continuous, hence continuous.

    Plato is telling you to pick that \delta to show the uniform continuity.
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