Geodesic
Best approximations by rational functions with respect to the Hausdorff distance
Matematičeskie zametki, Tome 11 (1972) no. 5, pp. 491-498 Cet article a éte moissonné depuis la source Math-Net.Ru

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Inverse theorems on the best approximations of plane sets in a Hausdorff metric by means of rational functions are cited. It is shown, among other things, that if $R_{n,r}(F,[a,b])=o(1/n)$, then there exists a set $P\subset[a,b]$ of complete measure over which $F$ constitutes a single-valued function. 
@article{MZM_1972_11_5_a2,
     author = {K. N. Lungu},
     title = {Best approximations by rational functions with respect to the {Hausdorff} distance},
     journal = {Matemati\v{c}eskie zametki},
     pages = {491--498},
     year = {1972},
     volume = {11},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_11_5_a2/}
}
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K. N. Lungu. Best approximations by rational functions with respect to the Hausdorff distance. Matematičeskie zametki, Tome 11 (1972) no. 5, pp. 491-498. http://geodesic.mathdoc.fr/item/MZM_1972_11_5_a2/