The statement "$X_n$ converges to $X$ in distribution", is equivalent to
A. $\limsup P[X_n< x]\leq P[X<x]$ for all real $x$.
B. $\liminf P[X_n< x]\geq P[X<x]$ and $\liminf P[X_n> x]\geq P[X>x]$ for all real $x$.
C. $E[g(X_n)]\to E[g(X)]$ for all bounded continuous functions $g$.
D. $E[g(X_n)]\to E[g(X)]$ for all uniformly continuous functions $g$.
I know that A and C are true. But I don't know how B and D both can be true too. Especially I never knew, if there is a theorem of D kind.
It would be very helpful if anyone give a reason or the specific theorem I need to know.