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MR1344203 (96j:49012) 49J45 (46A55 46B20 46N10 52A41)
Borwein, Jonathan M.; Vanderwerff, Jon D. Epigraphical and uniform convergence of convex functions. Trans. Amer. Math. Soc. 348 (1996), no. 4, 1617–1631.
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