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

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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. 
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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     author = {Aczel, J.},
     title = {On {Strict} {Monotonicity} of {Continuous} {Solutions} of {Certain} {Types} of {Functional} {Equations}},
     journal = {Canadian mathematical bulletin},
     pages = {229--232},
     year = {1966},
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     number = {2},
     doi = {10.4153/CMB-1966-031-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-031-6/}
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