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true/false:-

  1. There is a continuous onto function from the unit sphere in $\mathbb{R}^3$ to the complex plane $\mathbb{C}$.

  2. There is a non-constant continuous function from the open unit disc $D = \{z ∈ \mathbb{C} \mid |z| < 1 \}$ to $\mathbb{R}$ which takes only irrational values .

  3. $f \colon \mathbb{C} \to \mathbb{C}$ is an entire function such that the function $g(z)$ given by $g(z) = f( 1/z)$ has a pole at 0. Then f is a surjective map.

please help anyone.

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    That's three questions … Some hints: 1. compactness, 2. connectedness, 3. polynomial.2012-09-19
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    i did not understand the hint of the third problem. please explain briefly.thank you2012-09-19
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    @mintu: Look at something like $f(z)=e^z$, so that $g$ has an essential singularity at $z=0$. Recall that a function has an essential singularity iff it has infinitely many $z^{-1}$ power terms. In other words, for which $f$ can $g$ *really* have a pole at 0?2012-09-19

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