Some inverse limits of Cuntz algebras

We construct a nontrivial inverse system of Cuntz algebras ${{\cal O}{n}:2\leq n<\infty}$,whoseinverselimitis\ast -isomorphiconto${\cal O}{\infty}$\mathrm{.}Byusingthisresult,itisshownthatthe$K_{0}$-functor is discontinuous with respect to the inverse limit even if the limit is a C${}^{\ast }$-algebra.