continuity of a map

Question

let B be the closed unit ball & D the open unit ball. If g is a continuous function from B$\rightarrow R$ can one find always a continuous function from $R^2 \rightarrow R$ such that f=g on B?The same question applies to D.

Answer 1

For the closed ball, think about extending $g$ "radially" when $\|x\| \ge 1$ (there is no need to use complicated theorems about the normality of $\mathbb{R}^2$): $$f(x) = \begin{cases} g(x) & x \|x\| \le 1 \\ g(\frac{x}{\|x\|}) & \|x\| > 1 \end{cases}$$ (make sure to understand why this is continuous)

For the open ball, find a function that blows up near the boundary of $D$, for example $\frac{1}{1-\|x\|}$.