Geodesic
On Strict Monotonicity of Continuous Solutions of Certain Types of Functional Equations
Canadian mathematical bulletin, Tome 9 (1966) no. 2, pp. 229-232

Voir la notice de l'article provenant de la source Cambridge University Press

It is a commonplace that F is continuous on the cartesian square of the range of f if f is continuous and satisfies 1 say, for all real x, y (cf. e.g. [2]). A.D. Wallace has kindly called my attention to the fact, that this is trivial only if f is (constant or) strictly monotonic and asked for a simple proof of the strict monotonicity of f. The following could serve as such: if on an interval f is continuous, nonconstant and satisfies (1), then f is strictly monotonic there. 
Aczel, J. On Strict Monotonicity of Continuous Solutions of Certain Types of Functional Equations. Canadian mathematical bulletin, Tome 9 (1966) no. 2, pp. 229-232. doi: 10.4153/CMB-1966-031-6
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[1] 1. Aczél, J., Beiträge zur Theorie der geometrischen Objekte. III-IV. Acta Math. Acad. Sci. Hung. 8, (1957) 19-52; esp. pp. 45-47. Google Scholar

[2] 2. Aczél, J., Vorlesungen über Funktionalgleichungen and ihre Anwendungen, Birkhäuser-Verlag, Basel-Stuttgart 1961. 2nd, English edition: Lectures on Functional Equations and their Applications, Academic Press, New York- London, 1966; esp. Sects. 1.1.3, 2.2.2, 2.5.1. Google Scholar

[3] 3. Aczél, J., Kalmàr, L. and Mikusinski, J., Sur l'équation de translation, Studia Math. 12 (1951), 112-116. Google Scholar

[4] 4. Coifman, R., p-groupes de transformations contractantes et iteration continue des fonctions réelles. Thèse, Genève, 1965. Google Scholar

[5] 5. Golab, S. and Schinzel, A., Sur l'équation fonctionnelle f(x +yf(x)) = f(x)f(y). Publ. Math. Debrecen 6 (1959), 113-125. Google Scholar

[6] 6. Michel, H., Űber Monotonisierbarkeit von Iterationsgruppen reeller Funktionen. Publ. Math. Debrecen 9, (1962), 298-306. Google Scholar

[7] 7. Pólya, G. and Szegő, G., Aufgaben and Lehrsätze aus der Analysis. I. Springer-Verlag, Berlin, 1925. 2nd edition: Berlin-Gottingen-Heidelberg 1954, esp. p. VI. Google Scholar

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