Semicontinuity of a supremum of continuous functions

Detailed Solution

The pointwise supremum of any family of lower semicontinuous functions is always lower semicontinuous, and every continuous function is in particular lower semicontinuous. Hence $f=\sup_n f_n$ is lower semicontinuous on $\mathbb R$. This need not be full continuity: a standard example is $f_n(x)=\min(1,\max(0,nx))$, whose supremum is the step function equal to $0$ for $x\le0$ and $1$ for $x>0$, which jumps up at $x=0$ and is therefore not continuous and not upper semicontinuous there (it is, however, lower semicontinuous at the jump, consistent with the general result). So (A) and (C) can fail. Since lower semicontinuity is guaranteed (a stronger, more specific fact than mere measurability), option (D) understates what can be said. Thus the correct option is $\boxed{\mathrm{B}}$.