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The composition of two derivatives has a fixed point. (English) Zbl 1011.26004

Summary: We show that if \(f,g: [0,1]\to [0,1]\) are both Darboux Baire-1 functions, then their composition, \(f\circ g\), possesses a fixed point.

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
26A21 Classification of real functions; Baire classification of sets and functions
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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