Orthogonality and Additivity Modulo Z

Let E be a real inner product space of dimension at least 2. Suppose ƒ : E → ℝ satisfies Open image in new window If there exist a neighbourhood U of the origin and γ ∈ (0, 1/4) such that ƒ(U) ⊂ (−γ,γ) + ℤ, then there exist a real constant c and a continuous linear functional g : E → ℝ such that Open image in new window