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Math Help - function continuous at a point

  1. #1
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    function continuous at a point

    f:R-->R
    f(x)=1 if x = 0
    f(x)=0 otherwise.
    (a) Is f continuous at the point a = 0?
    (b) Is f continuous at the point a = 1?

    have i done them correctly?

    Continuous at a=0 because \lim_{x\to0}f(x)=1=f(0) correct?

    also continuous at a=1 because \lim_{x\to1}f(x)=0=f(1)

    thanks
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  2. #2
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    Quote Originally Posted by charikaar View Post
    f:R-->R
    f(x)=1 if x = 0
    f(x)=0 otherwise.
    (a) Is f continuous at the point a = 0?
    (b) Is f continuous at the point a = 1?

    have i done them correctly?

    Continuous at a=0 because \lim_{x\to0}f(x)=1=f(0) correct?

    also continuous at a=1 because \lim_{x\to1}f(x)=0=f(1)

    thanks
    Are you really sure of the first case? When you are not at 0 (which you aren't when you are taking a limit), the function takes the value 0. What does x_n=0 converge to?

    I mean you can think about it more intuitively, is the function "connected" at 0? Draw it.
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  3. #3
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    I see what you mean. The function is not connected at zero and hence the limit in first case doesn't exist so not continuous at a=0
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  4. #4
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    Quote Originally Posted by charikaar View Post
    I see what you mean. The function is not connected at zero and hence the limit in first case doesn't exist so not continuous at a=0
    The limit does exist! Try to compute it this way
    <br />
f(1/2), f(1/4), f(1/8),...<br />
    what it the limit of this sequence? (it's the same as \lim_{x \downarrow 0} f(x)).

    I think it may help if you look up the definition of a limit. The limit at a=0 might not be f(a) (which is the case here), but indeed it still does exist.
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