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On Strict Monotonicity of Continuous Solutions of Certain Types of Functional Equations

Published online by Cambridge University Press:  20 November 2018

J. Aczel
J. Aczel*
Affiliation:
University of Waterloo
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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.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 2 , June 1966 , pp. 229 - 232
Copyright
Copyright © Canadian Mathematical Society 1966

References

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