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Nov 22, 2023 at 16:12 comment added Iosif Pinelis @Akira : As in all such problems, the continuity plays here no role at all. Indeed, you can smooth out $\rho$ and $\rho_n$ in neighborhoods of the points of discontinuity so that $H(\rho)$ and $H(\rho_n)$ be approximated however closely.
Nov 22, 2023 at 16:07 vote accept Akira
Nov 22, 2023 at 15:56 comment added Akira It seems your $\rho$ is not continuous.
Nov 22, 2023 at 15:46 history answered Iosif Pinelis CC BY-SA 4.0