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Math Help - continuity of a function at a point

  1. #1
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    continuity of a function at a point

    Which of the following functions f are continuous at the point a=3 and a=2.5:

    f(x)=x if x is an integer.
    f(x)=0 if x is not an integer.

    I know the function is not continuous at a=3 and is continuous at a=2.5 but how do i prove this using Cauchy definition (epsilon-delta) of continuous functions.

    Thank you.
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  2. #2
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    Quote Originally Posted by charikaar View Post
    Which of the following functions f are continuous at the point a=3 and a=2.5:

    f(x)=x if x is an integer.
    f(x)=0 if x is not an integer.

    I know the function is not continuous at a=3 and is continuous at a=2.5 but how do i prove this using Cauchy definition (epsilon-delta) of continuous functions.

    Thank you.
    x=3:
    If I give you a y some small distance (less than 1) away from 3, you get |f(x)-f(y)|=3. So if you pick epsilon to be 2.5, then can you find any interval that will get me |f(x)-f(y)|<\epsilon?

    x=2.5:
    Pick delta to be 0.4, what do you get?
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    Thank you for your help. I can now solve a=2.5 but still struggle with a=3 bit ..in our notes |f(a+h)-f(a)|<\epsilon whats f(y) in your answer? thanks
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  4. #4
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    Quote Originally Posted by charikaar View Post
    Thank you for your help. I can now solve a=2.5 but still struggle with a=3 bit ..in our notes |f(a+h)-f(a)|<\epsilon whats f(y) in your answer? thanks
    The usual definition of continuity is; f is continuous at a point x if  \forall \epsilon >0, \exists \delta >0 \mbox{  s.t.  } |x-y|<\delta \implies |f(x)-f(y)|<\epsilon. If you want to use your definition, think about what happens if 0<h<1. You don't need to worry about large distances away from a (if you prove it for small h, then if 0<h<"a number bigger than 1", then you pick h<1).
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