# Extending a continuous function on the unit sphere to the unit ball

I'm trying to solve a problem in Lee's topology book. In this book, a closed n-cell $D$ is any space homeomorphic to the closed unit ball. Then $\mathrm{Int} D$ , resp. $\partial D$ are the images of the open unit ball, resp. the boundary of the closed unit ball under such a homeomorphism. Given a continuous function $f: \partial D \rightarrow \left [ 0,1 \right ]$, he wants one to show that $f$ extends to a continuous function $F: D \rightarrow \left [ 0,1 \right ]$ that is strictly positive in $\mathrm{Int} D$. Obviously, it suffices to show this for a closed unit ball and I was thinking something like $$\left \|x \right \|f\left(\frac{x}{\left \| x \right \|}\right) + \dfrac{1 -\left \| x \right \|}{2},$$ but that does not quite do it. Tietze's extension theorem is not in that book (and in fact is a weaker result). Any ideas? Thanks in advance.

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Which of his books? Intro to topological manifolds? – Rudy the Reindeer Jan 17 '13 at 20:40
Why does the map not do it? – Stefan Hamcke Jan 17 '13 at 22:08
I agree with Stefan H., your proposed $F$ works. – Matemáticos Chibchas Jan 17 '13 at 22:22
Dear @user58664, I apologize for the rubbish answer I just deleted. It was late when I wrote it and it adds absolutely nothing to your already fine answer. – Rudy the Reindeer Jan 18 '13 at 19:36