differentiable point set

**Shanks** · December 23rd 2009, 02:08 AM

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.

**Jose27** · December 23rd 2009, 09:26 AM

Originally Posted by Shanks

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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