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Math Help - Continuity

  1. #1
    Newbie
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    Oct 2008
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    Continuity

    Using the definition of continuity directly prove that f
    : (0,infinity) -->R defined by

    f
    (x) := 1/x is continuous.

    I know you have to start with epislon>0 and finding a delta, but I can't come up with the right delta. Help?
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  2. #2
    Junior Member
    Joined
    Aug 2010
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    easier way

    instead you could use the other characterization of continuity.
    \lim_{x\to c} f(x)   =    f(c)
    you kno for x not equal to zero 1/x is exist in R.
    now use algebraic limit theorem about reciprocal limits.
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  3. #3
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    Notice |\frac{1}{x}-\frac{1}{x_0}|= \frac{|x-x_0|}{xx_0}< \frac{\delta}{xx_0} <\frac{2\delta}{x_0^2} <\varepsilon provided \delta < \min \{ \frac{x_0}{2} , \frac{\varepsilon x_0^2}{2} \}
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