Density of __C__^∞^(Ω) in__W__^1,__p__ (__x__ )^(Ω) with discontinuous exponent __p__ (__x__ )

## Abstract In this paper, we present a sufficient condition for the density of __C__ ^∞^(Ω) in__W__ ^1,__p__ (__x__ )^(Ω) with __p__ (__x__ ) being discontinuous. Roughly speaking, we can get the density under the following assumptions: 1) Ω can be divided into some open pieces Ω~__i__~ such that __p__ (__x__ ) is “fine”on each Ω~__i__~ , for example, __p__ (__x__ ) satisfies the Dini–Lipschitz condition on each Ω~__i__~ . 2) The corresponding pieces “get along with each other regularly ”. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)