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How to construct the homeomorphism between $C[0,1]$ and $C[0,1]\setminus\{\theta\}$ in the explicit form?

Here, as usual, $C[0,1]$ is the Banach space of continuous functions $f:[0,1]\to\mathbb{R}$ with the norm $$ \|f\|=\max\limits_{x\in[0,1]}|f(x)|; $$ and $\theta(x)\equiv 0$ $\forall x\in[0,1]$.

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Is this a homework question? – Noah Schweber Jun 15 '13 at 23:15
This does not appear to be a homework question unless I am missing something. – Joseph Van Name Jun 16 '13 at 2:29
I would try to adapt the construction for $\ell_2$ to $c_0$ and use the fact that $c_0$ is isometric to a norm one complemented subspace of $C[0,1])$ in a natural way. – Bill Johnson Jun 16 '13 at 5:07
same question on MSE with helpful comment by Martin – Norbert Aug 19 '13 at 20:42

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