Let $f\colon\overline{\mathbb{D}}\to\mathbb{C}$ be a continuous function but that $f\colon\mathbb{D}\to\mathbb{C}$ is holomorphic. My question is

Can the restriction of $f$ to $\mathbb{S}$ assume its values in the unit interval $[0,1]$, that is $f(\mathbb{S})\subseteq[0,1]$?

Specific/explicit examples — if any — will be helpful.

($\mathbb{D}$ is the open unit disc with $\mathbb{S}$ as its boundary).