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Math Help - R->R function in C∞ whose set of critical points are dense

  1. #1
    Newbie ajskim's Avatar
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    R->R function in C∞ whose set of critical points are dense

    I need to find such a function. Any ideas about how to approach this? We have just covered Sard's theorem if that provides some sort of insight on how I should tackle this.
    Thanks
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    Quote Originally Posted by ajskim View Post
    I need to find such a function. Any ideas about how to approach this? We have just covered Sard's theorem if that provides some sort of insight on how I should tackle this.
    Thanks

    Sard's theorem is trivial for one variable functions, as f(X)=\{0\}\,,\,\,X= the set of critical points of f ,

    so I don't think it'll help a lot here.

    Of course, f(x)=c= constant is a trivial example, but I'm not sure how to construct other examples, or even whether there exist...

    Tonio
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    I was hoping to find a non-trivial example, but thank you. Yeah, I agree that Sard's probably isn't applicable here.
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  4. #4
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    Oops! I meant critical values, not critical points
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    Solved! Sum of bump functions
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