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



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> Can the restriction of $f$ to $\mathbb{S}$ assume its values in the unit interval $[0,1]$, that is $f(\mathbb{S})\subseteq[0,1]$?
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Specific/explicit examples — if any — will be helpful.


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



A boundary behaviour of holomorphic functions
As Dan Petersen anticipated, any function satisfying the requirements is constant. Indeed, the function defined as $g(z):=f(z)$ for $|z|\leq 1$ and $g(z):=\overline{f(1/\overline{z})}$ for $|z|\geq 1$ is entire and bounded, hence constant by Liouville's theorem.


