This paper addresses the problem of data compression with local decoding and local update. A compression scheme has worst-case local decoding $d_{wc}$ if any bit of the raw file can be recovered by probing at most $d_{wc}$ bits of the compressed sequence, and has update efficiency of $u_{wc}$ if a single bit of the raw file can be updated by modifying at most $u_{wc}$ bits of the compressed sequence. This article provides an entropy-achieving compression scheme for memoryless sources that simultaneously achieves $ O(\log\log n) $ local decoding and update efficiency. Key to this achievability result is a novel succinct data structure for sparse sequences which allows efficient local decoding and local update. Under general assumptions on the local decoder and update algorithms, a converse result shows that $d_{wc}$ and $u_{wc}$ must grow as $ Ω(\log\log n) $.