LetLϕ,λ = {ω ∈ Σ∗| ϕ(ω) > λ} be thelanguage recognized by a formal seriesϕ:Σ∗ → ℝ with isolated cut pointλ. We provide new conditions that guarantee the regularity of thelanguage Lϕ,λ in the case thatϕ is rational or ϕ is a Hadamard quotient of rationalseries. Moreover the decidability property of such conditions is investigated.