Geodesic
Level sets of continuous functions increasing with respect to each variable
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 25 (2005) no. 1, pp. 19-26.

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We are going to prove that level sets of continuous functions increasing with respect to each variable are arcwise connected (Theorem 3) and characterize those of them which are arcs (Theorem 2). In [3], we will apply the second result to the classical linear functional equation
Keywords: level set, continuous function, function increasing with respect to each variable, arcwise connectedness 
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Sajbura, Katarzyna. Level sets of continuous functions increasing with respect to each variable. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 25 (2005) no. 1, pp. 19-26. http://geodesic.mathdoc.fr/item/DMDICO_2005_25_1_a1/

[1] M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warszawa 1968.

[2] M. Kuczma, B. Choczewski and R. Ger, Iterative functional equations, Encyclopedia of mathematics and its applications 32, Cambridge University Press, Cambridge, 1990.

[3] K. Sajbura, On a linear functional equation with a mean-type mapping having no fixed points, Discuss. Math. DICO 25 (2005), 27-46.