# Is there a holomorphic function on the unit disk that satisfies a certain condition?

Is there a holomorphic function on the unit disk that satisfies $f(1/n) = 1/\sqrt{n}$?

Thougthts so far: I know that $f(z) = \sqrt{z}$ won't work, as it is not analytic at $0$. My intuition says that this is impossible. Just a hint at this point would be most helpful, as I am preparing for a qual.

• If such an $f$ existed, it would in particular be complex differentiable at $0$. – Daniel Fischer Aug 26 '15 at 19:00
• Way too complicated. What about difference quotients? – Daniel Fischer Aug 26 '15 at 19:16

If such a function existed, by continuity, we'd have $f(0) = 0$. Thus we have the difference quotients
$$\frac{f\bigl(\frac{1}{n}\bigr) - f(0)}{\frac{1}{n} - 0} = \frac{1/\sqrt{n}}{1/n} = \sqrt{n},$$
so such a function cannot be differentiable at $0$.
• I see how this can be used to show that $f'(0) = 0$, but why does that mean it can't be differentiable at $0$? – user19817 Aug 26 '15 at 19:27
• How do you get $f'(0) = 0$ from that? The sequence of difference quotients doesn't converge to $0$. – Daniel Fischer Aug 26 '15 at 19:29
• Ah I see, the $1/n$ tripped me up, I was thinking that $n \to 0$. You're right, this is simpler. – user19817 Aug 26 '15 at 19:30