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Math Help - continuous functions

  1. #1
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    continuous functions

    If f: [0,1] is continuous and has only rational [respectively, irrational] values, must f be constant? Prove your answer.
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  2. #2
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    does anyone have any idea how to do this??
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  3. #3
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    Quote Originally Posted by rmpatel5 View Post
    If f: [0,1] is continuous and has only rational [respectively, irrational] values, must f be constant? Prove your answer.
    Letís assume \left( {\forall x \in \left[ {0,1} \right]} \right)\left[ {f(x) \in \mathbb{Q}} \right] and suppose that \left( {\exists \left\{ {a,b} \right\} \subseteq \left[ {0,1} \right]} \right)\left[ {f(a) \ne f(b)} \right].
    Because the irrationals are dense there is an irrational number \gamma between f(a)\;\&\; f(b) but by the intermediate value theorem we know that {\exists c} between a\;\&\; b such that f(c)=\gamma.
    Do you see a contradiction?
    How does that imply the f is constant?
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  4. #4
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    Quote Originally Posted by Plato View Post
    Letís assume \left( {\forall x \in \left[ {0,1} \right]} \right)\left[ {f(x) \in \mathbb{Q}} \right] and suppose that \left( {\exists \left\{ {a,b} \right\} \subseteq \left[ {0,1} \right]} \right)\left[ {f(a) \ne f(b)} \right].
    Because the irrationals are dense there is an irrational number \gamma between f(a)\;\&\; f(b) but by the intermediate value theorem we know that {\exists c} between a\;\&\; b such that f(c)=\gamma.
    Do you see a contradiction?
    How does that imply the f is constant?
    I really dont see it. I have been trying to do this for the past couple hours i just dont understand this material. I hate real analysis
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  5. #5
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    Quote Originally Posted by rmpatel5 View Post
    I really dont see it. I have been trying to do this for the past couple hours i just dont understand this material. I hate real analysis
    Have you considered dropping the course and changing you major?
    That is the advice I have give many, many a student in my years as a department chairperson.
    To you own self be true. If this basic material is a real problem for you, I can tell you it only gets worst.
    Donít make the mistake of waiting too late to change course.
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