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Is there is a real differentiable function for which the graph of its derivative function is not topologically connected? Thanks a lot.

mathfan9
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  • Possible duplicate of Discontinuous derivative. –  Nov 09 '18 at 19:34
  • @ToposLogos: Possible duplicate of Discontinuous derivative --- I don't see the connection (pun intended). Everywhere continuous functions have connected graphs, and there exist nowhere continuous functions (even worse, functions whose graphs are dense in the plane) that have connected graphs. Regarding nowhere continuous examples, there's a well-known 1942 paper by F. B. Jones that shows the graphs of certain discontinuous additive functions (nowhere continuous, not Lebesgue measurable, have dense graphs, etc.) can be connected. – Dave L. Renfro Nov 09 '18 at 19:44

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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.

Dave L. Renfro
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  • Thank you for your excellent detailed answer. – mathfan9 Nov 10 '18 at 00:04
  • Could you improve your question as suggested, such as explaining the origin of the question for you? Incidentally (for others here), I'd be quite surprised if this was a homework problem or a research problem posed to a student because I suspect pretty much anyone who has the background to work on this would likely be aware of the result --- it's a fairly commonly known property (having a connected graph) of Baire $1$ Darboux functions, and anyone familiar with the combined property "Baire $1$ and Darboux" who would be considering this property for derivatives would likely know the answer. – Dave L. Renfro Nov 10 '18 at 09:11