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Math Help - differentiable point set

  1. #1
    Senior Member Shanks's Avatar
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    differentiable point set

    Find a suffice and neccessary condition that a set E in real numbers such that there is a real function whose differentiable point set is exactly E.
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  2. #2
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    Quote Originally Posted by Shanks View Post
    Find a suffice and neccessary condition that a set E in real numbers such that there is a real function whose differentiable point set is exactly E.
    Since we're talking about differentiability we have that E is open (or are you not allowed to assume this?). If E is open it is the union of disjoint open intervals so in each interval (a,b) you define f as follows: If z\in (a,b) then f(z)= \sum_{n=1}^{\infty } \left( \frac{z-z_0}{c} \right) ^n where z_0= a+\frac{b-a}{2} and c is such that \vert \frac{z-z_0}{c} \vert < 1. Then f is defined in every interval and cannot be extended beyond them.

    If you're not allowed to assume E open I don't really know how you could approach this...

    On a related note, this reminded me of this Domain of holomorphy - Wikipedia, the free encyclopedia in the case of one variable
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