Would you correct me? Which of the following statements are true?

a. There exists an entire function $f : \mathbb{C}\rightarrow\mathbb{C}$ which takes only real values. and is such that $f(0) = 0$ and $f(1) = 1$

b. There exists an entire function $f : \mathbb{C}\rightarrow\mathbb{C}$ such that $f(n + 1/n) = 0$ for all $n\in\mathbb{N}$

c. There exists an entire function $f : \mathbb{C}\rightarrow\mathbb{C}$ which is onto and which is such that $f(1/n) = 0$ for all $n\in\mathbb{N}$

Well, for a) it is only constants.

b) can have such non-constants entire function.

c) only constants.

Are my answers correct?plz help.

  • 1
    $\begingroup$ They are all correct if you remove the word "onto" from c (constant functions don't satisfy that). $\endgroup$ – Matt Apr 26 '12 at 21:34
  • $\begingroup$ For (a) if $c$ is a constant, then how can you have $c=0$ and $c=1$? Similarly for (c). Ironically, (b) is the only one for which there is a constant solution for $f$. (But I doubt that all solutions are constant) $\endgroup$ – Hurkyl Apr 26 '12 at 21:34

a. Not true because of the open mapping theorem. No nonconstant function can take on only real values, regardless of the other condition in the problem. Note that a constant function $f$ cannot satisfy $f(0)=0$ and $f(1)=1$, so we have to rule out constant solutions too, for trivial reasons.

b. You can construct a function with zeroes only at these points using the Weierstrass Factorization Theorem.

c. The zeros of $f$ have an accumulation point, so such an $f$ must be 0 everywhere.


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