If $u_n:\Bbb C\to \Bbb R$ are subharmonic, bounded, continuous, and $u_{n+1}\le u_n$ for all $n$, can their limit be discontinuous?
By boundedness, it is easy to show that the limit function $u$ will satisfy the submean value property using the MCT or DCT. Proving upper semicontinuity is also simple by just taking a union of open sets for the preimage $\{u<a\}$. Yet I can neither prove that $u$ will be continuous nor find a counterexample.
Intuitively I would be inclined to think that no counterexample can exist, because subharmonic functions are supposed to behave like convex functions. The limit of convex functions is convex, hence continuous, and Dini's theorem proves the result on compact sets. However I have no way of making this rigorous, so I am wondering if this even holds.