Geodesic
Continuous traces of $\delta$-subharmonic functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 40, Tome 401 (2012), pp. 189-197

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We consider the set of $\delta$-subharmonic functions with compact support in the plane. It is proved that the set of their traces on a segment does not include all continuous functions. 
@article{ZNSL_2012_401_a9,
     author = {K. A. Sukhov},
     title = {Continuous traces of $\delta$-subharmonic functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {189--197},
     publisher = {mathdoc},
     volume = {401},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a9/}
}
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K. A. Sukhov. Continuous traces of $\delta$-subharmonic functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 40, Tome 401 (2012), pp. 189-197. http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a9/