Is there is a real differentiable function for which the graph of its derivative function is not topologically connected? Thanks a lot.
2026-03-28 18:14:15.1774721655
A question on topology
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Every derivative has a connected graph. This follows from the more general fact that the graph of any Baire $1$ Darboux function is connected, which was first proved in [1] (see also Theorem 4.1 on p. 99 in [2], AND Theorem 7 at the top of p. 130 in [3], AND Theorem B on p. 182 in [4]). The actual observation that, since derivatives are Baire $1$ Darboux functions (Darboux himself showed “Darboux” in 1875 and Baire showed “Baire $1$” in his 1899 Ph.D. dissertation), it follows that graphs of derivatives are topologically connected in the plane, was first made in [5].
[1] Kazimierz [Casimir] Kuratowski and Wacław Sierpiński, Les fonctions de classe $1$ et les ensembles connexes punctiformes [On functions of class $1$ and connected punctiforme sets], Fundamenta Mathematicae 3 (1922), 303-313.
[2] Andrew Michael Bruckner and Jack Gary Ceder, Darboux continuity, Jahresbericht der Deutschen Mathematiker-Vereinigung 67 (1965), 93-117.
[3] Kazimierz Kuratowski, Topology, Volume II, Academic Press, 1968.
[4] Jack Gary Cedar and Terrance Laverne Pearson, A survey of Darboux Baire $1$ functions, Real Analysis Exchange 9 #1 (1983-1984), 179-194.
[5] Bronisław Knaster and Kazimierz [Casimir] Kuratowski, Sur quelques propriétés topologiques des fonctions dérivées [On some topological properties of derivative functions], Rendiconti del Circolo Matematico di Palermo (1) 49 (1925), 382-386.