Geodesic
Denjoy--Khinchin-integrable functions and their conjugates
Izvestiya. Mathematics , Tome 11 (1977) no. 3, pp. 625-664.

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In this paper $2\pi$-periodic functions $\Phi(x)$ and $\Psi(x)$ are constructed so that they are both Denjoy–Khinchin integrable, are not equivalent to zero, and have conjugates $\overline\Phi$ and $\overline\Psi$ satisfying $\overline\Phi(x)=0$ almost everywhere and $\overline\Psi(x)=1$ almost everywhere. Bibliography: 12 titles. 
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T. P. Lukashenko. Denjoy--Khinchin-integrable functions and their conjugates. Izvestiya. Mathematics , Tome 11 (1977) no. 3, pp. 625-664. http://geodesic.mathdoc.fr/item/IM2_1977_11_3_a8/

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[9] Lukashenko T. P., “O funktsiyakh, sopryazhennykh s $D-X$-integriruemymi funktsiyami”, Izv. AN SSSR. Ser. matem., 35 (1971), 381–407 | Zbl

[10] Lukashenko T. P., “Sopryazhennye funktsii i integraly”, Izv. AN SSSR. Ser. matem., 36 (1972), 435–449 | Zbl

[11] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, t. 1, Nauka, M., 1969

[12] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, t. 2, Nauka, M., 1969