# Continuous Extension of $S^{n-1} \to [0,b]$ to $B^n \to [0,b]$.

Let $f : S^{n-1} \to [0,b] \subset \mathbb{R}$ be a continuous function. Does there exist a continuous extension $F : B^n \to [0,b]$ of $f$ that is strictly positive on $\mathrm{Int} (B^n)$?

-
Well, not if $b=-1$. Did you mean to assume $a=0$? –  Chris Eagle Mar 5 '12 at 22:49
Yes, sorry, I forgot to specify. –  Rick Mar 5 '12 at 22:51
$$F(v) = \begin{cases} b & v=0 \\ \|v\|f(\frac{1}{\|v\|}v) + (1-\|v\|)b& v\ne 0 \end{cases}$$ should do it.