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The problem at hand is:

let $f$ be a nonconstant and entire function.

  1. Prove that for all $\omega \in \mathbb{C}$ and for all $\varepsilon > 0 $ there exists $z \in \mathbb{C}$ such that $|f(z) - w| < \varepsilon $

  2. Deduce that $f(\mathbb{C})$ is dense in $\mathbb{C}$

  3. Is $f$ necessarily onto?

We have not taken the Casorati-Weierstrass theorem yet but I figured this is a variation from it from what I've read on the net. I don't think I understand the approaches for proofs so far..

My attempt at 1 is:

$f$ is entire, therefore it is continuous.

Then for all $\varepsilon > 0$ there exists $\delta$ such that taking $z_1 \in \mathbb{C}$, $|z - z_1| < \delta$ implies $|f(z) - f(z_1)| < \varepsilon$

Therefore, letting $\omega = f(z_1)$ for any $z_1 \in \mathbb{C}$ we get the result that for any $\omega$ there exists a $z$ such that $|f(z) - w| < \varepsilon $

Is this approach okay? I did not use the fact that the function is nonconstant. How does that relate to density?

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    I don't think you can say "Let $\omega = f(z_1)$" in your proof because this potentially restricts the values of $\omega$. You must select $\omega$ arbitrarily from $\mathbb{C}$. I don't think we're allowed to know from the outset that the range of $f$ is the whole plane. – WB-man Mar 11 '17 at 19:29
  • Please make sure that the titles you pick for your questions at least make sense! A function cannot be dense — its image, on the other hand... – Mariano Suárez-Álvarez Mar 11 '17 at 19:36
  • but how doesn't it make sense?.. –  Mar 11 '17 at 19:39
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    Strongly related (duplicate?): http://math.stackexchange.com/questions/2170634/show-that-for-any-w-in-mathbbc-there-exists-a-sequence-z-n-s-t-fz-n. – Martin R Mar 11 '17 at 19:50
  • I have solved the first 2 parts...any hint for the third? –  Mar 11 '17 at 20:58
  • For 3rd, consider $f(z) = e^z$. – Ethan Alwaise Mar 13 '17 at 03:23

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