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Well, for (a) I have no idea how to extend, I feel that there will be a continuous extension. For example I can define $f=g$ when $f$ takes values from upper boundary of the disk and upper half plane and same way for lower boundary and lower half plane. For (b) I have no idea.

(c) is true by Uryshon's lemma? As the plane is normal, and the given sets are closed and disjoint.

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For (a), just define $f(x) = g(\frac{x}{\|x\|})$, for $x \notin B$. For (b),(c), it is simpler to look at functions $\mathbb{R} \to \mathbb{R}$ first. – copper.hat Jul 19 '12 at 21:12
up vote 3 down vote accepted

For (a) you can use Tietze's extension theorem:

$B$ is a closed subset of a normal space.

For (b) I think you could take any function with a discontinuity at distance one from the origin (like for example $\frac{1}{1 - |x^2 + y^2|}$)

For (c) your answer is correct: the first set is a circle around $0$ of radius $1.5$, the second set is the closed unit disk union the complement of an open disk of radius $2$ so that we have two closed disjoint sets and can apply Urysohn.

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