Geodesic
An irremovable Carleman singularity for Haar's system
Matematičeskie zametki, Tome 14 (1973) no. 6, pp. 799-807.

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An example is constructed of a continuous function $f(x)$ that has the property that any function in $L_{(01)}^2$ that coincides with $f(x)$ on a set of positive measure realizes a Carleman singularity for Haar's system. 
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     author = {Yu. S. Fridlyand},
     title = {An irremovable {Carleman} singularity for {Haar's} system},
     journal = {Matemati\v{c}eskie zametki},
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     year = {1973},
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Yu. S. Fridlyand. An irremovable Carleman singularity for Haar's system. Matematičeskie zametki, Tome 14 (1973) no. 6, pp. 799-807. http://geodesic.mathdoc.fr/item/MZM_1973_14_6_a3/