Geodesic
Some Remarks on Nonlinear Composition Operators in Spaces of Differentiable Functions
Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 3, pp. 321-336.

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In this note we study the nonlinear composition operator $f \mapsto g \circ f$ in various spaces of differentiable functions over an interval. It turns out that this operator is always bounded in the corresponding norm, whenever it maps such a space into itself, but continuous only in exceptional cases. 
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Appell, J.; Jesús, Z.; Mejía, O. Some Remarks on Nonlinear Composition Operators in Spaces of Differentiable Functions. Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 3, pp. 321-336. http://geodesic.mathdoc.fr/item/BUMI_2011_9_4_3_a1/

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