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It is shown for certain planar sets, e.g., Brennan sets, with no interior, that if      f ∈  R 2 ( X , d x d y ) f \in {R^2}(X,dxdy)       and      | f | = 1 a.e. d x d y |f| = 1\; \text {a.e.}\; dxdy       then      f ≡  constant f \equiv \text {constant}       .

On the nonexistence of unimodular functions in 𝑅²(𝑋,𝑑𝑥𝑑𝑦)