Geodesic
Continuity of operators in separable constructive metric spaces
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part VII, Tome 60 (1976), pp. 194-196

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We construct an everywhere defined continuous operator on a separable CMS having the following property: it is not possible to cover the domain of this operator by denumerably many balls such that the oscillation of the operator in each ball of the covering is less than 1. 
@article{ZNSL_1976_60_a13,
     author = {S. V. Pakhomov},
     title = {Continuity of operators in separable constructive metric spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {194--196},
     publisher = {mathdoc},
     volume = {60},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1976_60_a13/}
}
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S. V. Pakhomov. Continuity of operators in separable constructive metric spaces. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part VII, Tome 60 (1976), pp. 194-196. http://geodesic.mathdoc.fr/item/ZNSL_1976_60_a13/