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 $\mathrm{Int} (B^n)$?
Continuous Extension of $S^{n-1} \to [0,b]$ to $B^n \to [0,b]$.
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general-topology
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0Yes, sorry, I forgot to specify. – 2012-03-05
1 Answers
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$F(v) = \begin{cases} b & v=0 \\ \|v\|f(\frac{1}{\|v\|}v) + (1-\|v\|)b& v\ne 0 \end{cases}$ should do it.
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0I see. I was playing with similar formulas, but couldn't quite get it to work. Thank you! – 2012-03-05