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 $\theta(x)\equiv 0$ $\forall x\in[0,1]$.