An Access Theorem for Continuous Functions

Let f be a continuous function on an open subset Ω of R 2 such that for every x ∈ Ω there exists a continuous map γ : [−1, 1] → Ω with γ(0) = x and f ∘ γ increasing on [−1, 1]. Then for every γ ∈ Ω there exists a continuous map γ : [0, 1) → Ω such that γ(0) = y , f ∘ γ is increasing on [0; 1), and for every compact subset K of Ω, max{ t : γ( t ) ∈ K } < 1. This result gives an answer to a question posed by M. Ortel. Furthermore, an example shows that this result is not valid in higher dimensions.