a multiple choice question on continuity between two different spaces

Question

Let
$B$ = {$(x, y) ∈ \mathbb{R}^2 : x^2 + y^2 ≤1$} and $D$ = {$(x, y) ∈ \mathbb{R}^2 : x^2 + y^2 < 1$}.
Pick out the true statements.
(a) Given a continuous function $g : B → \mathbb{R}$, there always exists a continuous function $f : \mathbb{R}^2 → \mathbb{R}$ such that $f = g$ on $B$.
(b) Given a continuous function $g : D →\mathbb{R}$, there always exists a continuous function $f : \mathbb{R}^2 → \mathbb{R}$ such that $f = g$ on $D$.
(c) There exists a continous function $f : \mathbb{R}^2 → \mathbb{R}$ such that $f ≡1$ on the set {$(x, y) ∈ \mathbb{R}^2 : x^2+y^2 = 3/2$}
and $f ≡ 0$ on the set
$B$∪{$(x, y) ∈ \mathbb{R}^2 : x^2+y^2≥2$}.

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i am totally stuck on this problem.please help somebody.

Answer 1

For($a$), consider the Tietze Extension Theorem (and this also clues you in for ($b$). For ($c$), consider Urysohn's Lemma.