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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 $\mathrm{Int} (B^n)$?

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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
up vote 3 down vote accepted

$$F(v) = \begin{cases} b & v=0 \\ \|v\|f(\frac{1}{\|v\|}v) + (1-\|v\|)b& v\ne 0 \end{cases}$$ should do it.

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I see. I was playing with similar formulas, but couldn't quite get it to work. Thank you! – Rick Mar 5 '12 at 23:02

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