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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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  • $\begingroup$ Is this a homework question? $\endgroup$
    – Noah Schweber
    Commented Jun 15, 2013 at 23:15
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    $\begingroup$ This does not appear to be a homework question unless I am missing something. $\endgroup$
    – Joseph Van Name
    Commented Jun 16, 2013 at 2:29
  • $\begingroup$ 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. $\endgroup$
    – Bill Johnson
    Commented Jun 16, 2013 at 5:07
  • $\begingroup$ same question on MSE with helpful comment by Martin $\endgroup$
    – Norbert
    Commented Aug 19, 2013 at 20:42

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