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Math Help - continuity at a point

  1. #1
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    continuity at a point

    using the definition:  \forall \epsilon >0, \exists \delta >0, \forall h, \ |h|<\delta \ : |f(a+h)-f(a)|<\epsilon.

    f(x)=x if x is an integer.
    f(x)=0 if x is not an integer. Prove that f is not continuous at=3 so we use the negation of the above definition.

    how do we know h=\min(\delta/2,0.5) Is there a general method?

    Thanks
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  2. #2
    Senior Member
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    Let \epsilon<1. Then for all |h|<1, |f(3+h)-f(3)|=|0-3|=3>\epsilon. In this case, its probably easier to look at the one-sided limits at an integer: since f is 0 as you approach an integer, the limit of f at any integer is 0, but f is not zero there, so it is discontinuous.
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