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Let 〈hn〉 denote a sequence of positive real numbers. We show that for every set A ⊆ ℝ there exists a function f: ℝ → ω such that A = {x ∈ ℝ: (∈〈hn〉)[hn ↘ 0 &amp; (∀n ∈ ℕ)(f(x − hn) = f(x + hn) = f(x))]}. This solves a problem of K. Ciesielski, K. Muthuvel and A. Nowik.

Every set is “symmetric” for some function