three true/false questions

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.

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That's three questions … Some hints: 1. compactness, 2. connectedness, 3. polynomial. – Harald Hanche-Olsen Sep 19 '12 at 16:52
i did not understand the hint of the third problem. please explain briefly.thank you – mintu Sep 19 '12 at 17:17
@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? – Alex R. Sep 19 '12 at 18:10

(1) False. The continuous image of a compact space is compact, and a compact subset of the plane is bounded. (2) False. The continuous image of a connected set is connected, and the connected subsets of $\mathbb{R}$ are intervals. (3) True. An entire function with a pole and infinite is a polynomial. Now apply the fundamental theorem of algebra.
The image of a compact space under a continuous function is compact. That answers your first question if you know the plane is not compact and the sphere is. The sphere is a closed and bounded subset of a Euclidean space, so it's compact. It's closed because it's the inverse image of the closed set $\{1\}$ under the continuous function $(x,y,z)\mapsto x^2+y^2+z^2$.