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Math Help - How to check the continuity

  1. #1
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    How to check the continuity

    How to check the continuity of function
    f(x) = x when x is rational
    2-x when x is irrational
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    Lightbulb Re: How to check the continuity

    Any function y=f(x) is continuous at point x=a if the following three conditions are satisfied :

    1) f(a) is defined.

    2) \lim_{x \to a}f(x) exists i.e is finite.

    3) \lim_{x \to a}f(x)=f(a)
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  3. #3
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    Re: How to check the continuity

    Quote Originally Posted by deepak View Post
    How to check the continuity of function
    f(x) = x when x is rational
    2-x when x is irrational
    Math Tutor - Functions - Theory - Elementary Functions
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    Re: How to check the continuity

    And, of course, there exist both rational and irrational numbers arbitrarily close to any number. So you can take the limit at any number by a sequence of rational numbers or a sequence of irrational numbers. I think you will find that this function is continuous at exactly one value of x. What is that value of x?
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  5. #5
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    Re: How to check the continuity

    here is a hint: at any real number a, let ε = |1-a|. use the definition of continuity.

    (there is one point, at which you'll have to choose some other ε. why is this not a problem?).
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