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Which of the following statements are true?

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

  2. There exists an entire function $f : C\rightarrow C$ such that $f(n + {1\over n}) = 0$ for all positive integers $n$.

  3. There exists an entire function $f : C\rightarrow C$ which is onto and which is such that $f({1\over n}) = 0$ for all positive integers $n$.

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1  
What do you mean by $f(n+1n)$? –  Gerry Myerson Aug 2 '12 at 4:02
    
What have you tried? What relevant facts do you know? (For example, are you familiar with Picard's theorems?) –  Brad Aug 2 '12 at 5:17
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2 Answers

  1. No such function exists. By Little Picard's Theorem (see wikipedia), if $f$ is entire and non-constant, then $f$ can miss at most one value.

  2. Here I am assuming you want an entire function with zeros at $n + \frac{1}{n}$. Since $n + \frac{1}{n}$ does not have a limit point (goes to $\infty$), you can use the Hadamard product formula to produce an entire function with exactly those values as it zeros (and you can even control its order of growth).

  3. No such function exists. Since $(\frac{1}{n})_{n \in \omega}$ has a limit point in $\mathbb{C}$, by analytic continuation, this function must be the constant zero function.

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3  
Picard's theorem is overkill for this one. It's easy to prove that a real-valued analytic function is constant. –  Robert Israel Aug 2 '12 at 6:55
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To follow up on Robert's comment: just look at the Cauchy-Riemann equations. –  t.b. Aug 2 '12 at 7:23
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$1$ is false: Non-constant entire function takes every value in $\mathbb C$ with one possible exception.

$2$ is true: $f=0$.

$3$ is false: Here $f$ is a non-constant entire function & so $f$ takes every value in $\mathbb C$ with one possible exception. {$1\over n$} is a sequence of distinct points converging to $0\implies f=0$ $!$

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I like the proof for 2) :D –  CBenni Dec 16 '12 at 12:00
    
Beginners always try to go for trivial examples :) –  Sugata Adhya Dec 16 '12 at 12:03
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