A "holomorphic" Peano curve?

A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square.

I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous map $\phi$ from the unit disk $|z|\le 1$ to $\mathbb C^1$ such that $\phi$ is holomorphic for $|z|<1$ and the image of the boundary $|z|=1$ has non-empty interior in $\mathbb C^1$ under the map $\phi$.

Define $$\phi(z):=\frac{1}{2\pi i}\int_{S^1}(\zeta-z)^{-1}\cdot \varphi(\zeta)d\zeta$$
for $|z|<1$, where $\varphi: S^1\to\mathbb{C}$ is a Peano curve (i.e. its image has nonempty interior), and $\phi(z):=\varphi(z)$ for $z\in S^1$. [Edit: this construction doesn't work because $\phi$, as I defined it, may not be continuous up to the boundary - see the comments]
• In general there is no reason for this function $\phi$ to be contunuous on $\overline{\mathbb{D}}$! For example, if negative terms are present in the Fourier expansion of $g$, then certainly a continuous extension of the Cauchy Transform of $g$ to the circle will not be equal to $g$. Nov 29 '12 at 18:16