Geodesic
Existence of Certain Analytic Homeomorphisms
Canadian mathematical bulletin, Tome 2 (1959) no. 2, pp. 71-75

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This note has its origin in the following problem: do there exist non-trivial increasing continuous functions on [0, l] to [0, l], which map the following sets in [0, l] onto themselves: the rational, the algebraic and the transcendental numbers? One such function is obviously f(x) = x; more generally, f(x) = (c + l)x/(cx + 1), with c rational and non-negative, satisfies the conditions. Let G denote the space of order-preserving homeomorphisms of [0, l} onto [0, l], in the uniform metric. It follows from Theorem 1 below that the set S of all such functions i s dense in G. S is clearly a subgroup of G and one may ask what a r e its group-theoretic properties. We shall not consider these questions. 
Melzak, Z. A. Existence of Certain Analytic Homeomorphisms. Canadian mathematical bulletin, Tome 2 (1959) no. 2, pp. 71-75. doi: 10.4153/CMB-1959-010-8
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     title = {Existence of {Certain} {Analytic} {Homeomorphisms}},
     journal = {Canadian mathematical bulletin},
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     year = {1959},
     volume = {2},
     number = {2},
     doi = {10.4153/CMB-1959-010-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1959-010-8/}
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